{"id":2936,"date":"2025-06-06T06:56:42","date_gmt":"2025-06-06T06:56:42","guid":{"rendered":"https:\/\/diznr.com\/?p=2936"},"modified":"2025-06-06T06:56:42","modified_gmt":"2025-06-06T06:56:42","slug":"binary-operator-concept-gate-2021-discrete-mathematics-tutorial-in-hindi-the-operator-binary","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/binary-operator-concept-gate-2021-discrete-mathematics-tutorial-in-hindi-the-operator-binary\/","title":{"rendered":"Binary Operator concept- GATE 2025 Discrete mathematics tutorial in Hindi The binary operator"},"content":{"rendered":"<p>Binary Operator concept- GATE 2021 Discrete mathematics tutorial in Hindi The binary operator<\/p>\n<p>[fvplayer id=&#8221;178&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"87\"><strong data-start=\"4\" data-end=\"85\">\u00a0Binary Operator Concept &#8211; GATE 2025 Discrete Mathematics Tutorial in Hindi<\/strong><\/h3>\n<h4 data-start=\"89\" data-end=\"127\"><strong data-start=\"94\" data-end=\"125\">\u00a0\u0915\u094d\u092f\u093e \u0939\u0948 Binary Operator?<\/strong><\/h4>\n<p data-start=\"128\" data-end=\"279\">Binary Operator \u090f\u0915 \u0910\u0938\u093e <strong data-start=\"151\" data-end=\"161\">\u0911\u092a\u0930\u0947\u091f\u0930<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b <strong data-start=\"173\" data-end=\"206\">\u0926\u094b \u0911\u092a\u0930\u0947\u091f\u093f\u0902\u0917 \u0935\u0948\u0932\u094d\u092f\u0942 (Operands)<\/strong> \u092a\u0930 \u0915\u093e\u092e \u0915\u0930\u0924\u093e \u0939\u0948 \u0914\u0930 \u090f\u0915 \u0938\u093f\u0902\u0917\u0932 \u0930\u093f\u091c\u0932\u094d\u091f \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0947 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0932\u093f\u0916\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">A\u2217BA * B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"294\" data-end=\"386\">\u091c\u0939\u093e\u0901 <span class=\"katex\"><span class=\"katex-mathml\">\u2217*<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2217<\/span><\/span><\/span><\/span> \u090f\u0915 <strong data-start=\"310\" data-end=\"327\">\u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930<\/strong> \u0939\u0948 \u0914\u0930 <strong data-start=\"334\" data-end=\"357\"><span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> \u0924\u0925\u093e <span class=\"katex\"><span class=\"katex-mathml\">BB<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><\/strong> \u0907\u0938\u0915\u0947 <strong data-start=\"363\" data-end=\"379\">\u0926\u094b \u0911\u092a\u0930\u0947\u0923\u094d\u0921\u094d\u0938<\/strong> \u0939\u0948\u0902\u0964<\/p>\n<h3 data-start=\"393\" data-end=\"453\"><strong data-start=\"397\" data-end=\"453\">\u00a0Types of Binary Operators in Discrete Mathematics<\/strong><\/h3>\n<p data-start=\"454\" data-end=\"567\">Discrete Mathematics \u092e\u0947\u0902 <strong data-start=\"479\" data-end=\"498\">\u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930\u094d\u0938<\/strong> \u092e\u0941\u0916\u094d\u092f\u0924\u0903 <strong data-start=\"507\" data-end=\"541\">Set Theory, Logic, and Algebra<\/strong> \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u090f \u091c\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h3 data-start=\"569\" data-end=\"631\"><strong data-start=\"573\" data-end=\"631\">\u00a0Arithmetic Binary Operators (\u0917\u0923\u093f\u0924\u0940\u092f \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930)<\/strong><\/h3>\n<p data-start=\"632\" data-end=\"830\"><strong data-start=\"635\" data-end=\"651\">Addition (+)<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">a+ba + b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><br data-start=\"665\" data-end=\"668\" \/><strong data-start=\"671\" data-end=\"690\">Subtraction (-)<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">a\u2212ba &#8211; b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><br data-start=\"704\" data-end=\"707\" \/><strong data-start=\"710\" data-end=\"732\">Multiplication (\u00d7)<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">a\u00d7ba \\times b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><br data-start=\"751\" data-end=\"754\" \/><strong data-start=\"757\" data-end=\"773\">Division (\u00f7)<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">a\u00f7ba \\div b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u00f7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><br data-start=\"790\" data-end=\"793\" \/><strong data-start=\"796\" data-end=\"811\">Modulus (%)<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">amod\u2009\u2009ba \\mod b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathrm\">mod<\/span><\/span><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"832\" data-end=\"883\"><strong data-start=\"834\" data-end=\"845\">Example<\/strong>: \u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">a=5a = 5<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">b=3b = 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, \u0924\u094b<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">5+3=85 + 3 = 8<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"906\" data-end=\"966\"><strong data-start=\"910\" data-end=\"966\">\u00a0Logical Binary Operators (\u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930)<\/strong><\/h3>\n<p data-start=\"967\" data-end=\"1023\">Boolean Algebra \u0914\u0930 <strong data-start=\"986\" data-end=\"1001\">Logic Gates<\/strong> \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"1025\" data-end=\"1134\"><strong data-start=\"1028\" data-end=\"1041\">AND ( \u2227 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2227BA \\land B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1059\" data-end=\"1062\" \/><strong data-start=\"1065\" data-end=\"1077\">OR ( \u2228 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2228BA \\lor B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1094\" data-end=\"1097\" \/><strong data-start=\"1100\" data-end=\"1113\">XOR ( \u2295 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2295BA \\oplus B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2295<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1136\" data-end=\"1156\"><strong data-start=\"1138\" data-end=\"1149\">Example<\/strong>: \u092f\u0926\u093f<\/p>\n<ul data-start=\"1157\" data-end=\"1286\">\n<li data-start=\"1157\" data-end=\"1185\"><span class=\"katex\"><span class=\"katex-mathml\">A=1A = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">B=0B = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1186\" data-end=\"1237\"><span class=\"katex\"><span class=\"katex-mathml\">A\u2227B=1\u22270=0A \\land B = 1 \\land 0 = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span> (AND Operation)<\/li>\n<li data-start=\"1238\" data-end=\"1286\"><span class=\"katex\"><span class=\"katex-mathml\">A\u2228B=1\u22280=1A \\lor B = 1 \\lor 0 = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> (OR Operation)<\/li>\n<\/ul>\n<h3 data-start=\"1293\" data-end=\"1356\"><strong data-start=\"1297\" data-end=\"1356\">\u00a0Relational Binary Operators (\u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930)<\/strong><\/h3>\n<p data-start=\"1357\" data-end=\"1455\">\u092f\u0939 \u0911\u092a\u0930\u0947\u091f\u0930\u094d\u0938 <strong data-start=\"1369\" data-end=\"1392\">\u0938\u0902\u092c\u0902\u0927\u094b\u0902 (Relations)<\/strong> \u0914\u0930 <strong data-start=\"1396\" data-end=\"1421\">\u0938\u0924\u094d\u092f\u0924\u093e (Truth Values)<\/strong> \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"1457\" data-end=\"1620\"><strong data-start=\"1460\" data-end=\"1478\">Equal to ( = )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A=BA = B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1492\" data-end=\"1495\" \/><strong data-start=\"1498\" data-end=\"1520\">Not Equal to ( \u2260 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2260BA \\neq B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"inner\"><span class=\"mord\">\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1537\" data-end=\"1540\" \/><strong data-start=\"1543\" data-end=\"1565\">Greater than ( &gt; )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A&gt;BA &gt; B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1579\" data-end=\"1582\" \/><strong data-start=\"1585\" data-end=\"1604\">Less than ( &lt; )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A&lt;BA &lt; B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1622\" data-end=\"1667\"><strong data-start=\"1624\" data-end=\"1635\">Example<\/strong>:<br data-start=\"1636\" data-end=\"1639\" \/>\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">A=4,B=6A = 4, B = 6<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>, \u0924\u094b<\/p>\n<ul data-start=\"1668\" data-end=\"1714\">\n<li data-start=\"1668\" data-end=\"1690\"><span class=\"katex\"><span class=\"katex-mathml\">A&lt;BA &lt; B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> (True)<\/li>\n<li data-start=\"1691\" data-end=\"1714\"><span class=\"katex\"><span class=\"katex-mathml\">A=BA = B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> (False)<\/li>\n<\/ul>\n<h3 data-start=\"1721\" data-end=\"1793\"><strong data-start=\"1725\" data-end=\"1793\">\u00a0Set Theory Binary Operators (\u0938\u0947\u091f \u0925\u094d\u092f\u094b\u0930\u0940 \u092e\u0947\u0902 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930\u094d\u0938)<\/strong><\/h3>\n<p data-start=\"1794\" data-end=\"1884\">Set Theory \u092e\u0947\u0902 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u091f\u0930\u094d\u0938 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 <strong data-start=\"1834\" data-end=\"1856\">Sets \u0915\u0947 \u092c\u0940\u091a \u0911\u092a\u0930\u0947\u0936\u0928<\/strong> \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p data-start=\"1886\" data-end=\"2060\"><strong data-start=\"1889\" data-end=\"1904\">Union ( \u222a )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u222aBA \\cup B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u222a<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1921\" data-end=\"1924\" \/><strong data-start=\"1927\" data-end=\"1949\">Intersection ( \u2229 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2229BA \\cap B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"1966\" data-end=\"1969\" \/><strong data-start=\"1972\" data-end=\"1992\">Difference ( &#8211; )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u2212BA &#8211; B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><br data-start=\"2006\" data-end=\"2009\" \/><strong data-start=\"2012\" data-end=\"2039\">Cartesian Product ( \u00d7 )<\/strong> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">A\u00d7BA \\times B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2062\" data-end=\"2131\"><strong data-start=\"2064\" data-end=\"2075\">Example<\/strong>:<br data-start=\"2076\" data-end=\"2079\" \/>\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">A={1,2,3}A = \\{1,2,3\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">B={2,3,4}B = \\{2,3,4\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">4<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>, \u0924\u094b<\/p>\n<ul data-start=\"2132\" data-end=\"2193\">\n<li data-start=\"2132\" data-end=\"2164\"><span class=\"katex\"><span class=\"katex-mathml\">A\u222aB={1,2,3,4}A \\cup B = \\{1,2,3,4\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u222a<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">4<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"2165\" data-end=\"2193\"><span class=\"katex\"><span class=\"katex-mathml\">A\u2229B={2,3}A \\cap B = \\{2,3\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<h3 data-start=\"2200\" data-end=\"2251\"><strong data-start=\"2204\" data-end=\"2251\">\u00a0Important Properties of Binary Operators<\/strong><\/h3>\n<p data-start=\"2252\" data-end=\"2707\"><strong data-start=\"2254\" data-end=\"2275\">Closure Property:<\/strong> \u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">a,ba, b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> \u0915\u093f\u0938\u0940 Set <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u090f\u0932\u093f\u092e\u0947\u0902\u091f \u0939\u0948\u0902 \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">a\u2217ba * b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> \u092b\u093f\u0930 \u0938\u0947 <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u092e\u0947\u0902 \u0906\u0924\u093e \u0939\u0948, \u0924\u094b \u0911\u092a\u0930\u0947\u0936\u0928 <strong data-start=\"2375\" data-end=\"2386\">Closure<\/strong> \u0915\u094b \u092b\u0949\u0932\u094b \u0915\u0930\u0924\u093e \u0939\u0948\u0964<br data-start=\"2403\" data-end=\"2406\" \/><strong data-start=\"2408\" data-end=\"2426\">Associativity:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">(A\u2217B)\u2217C=A\u2217(B\u2217C)(A * B) * C = A * (B * C)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><br data-start=\"2458\" data-end=\"2461\" \/><strong data-start=\"2463\" data-end=\"2481\">Commutativity:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">A\u2217B=B\u2217AA * B = B * A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span><br data-start=\"2501\" data-end=\"2504\" \/><strong data-start=\"2506\" data-end=\"2527\">Identity Element:<\/strong> \u0910\u0938\u093e \u090f\u0932\u093f\u092e\u0947\u0902\u091f <span class=\"katex\"><span class=\"katex-mathml\">ee<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><\/span><\/span><\/span> \u091c\u093f\u0938\u0938\u0947 <span class=\"katex\"><span class=\"katex-mathml\">A\u2217e=AA * e = A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span> \u0939\u094b\u0964 (\u091c\u0948\u0938\u0947 Addition \u092e\u0947\u0902 0, Multiplication \u092e\u0947\u0902 1)<br data-start=\"2617\" data-end=\"2620\" \/><strong data-start=\"2622\" data-end=\"2642\">Inverse Element:<\/strong> \u0910\u0938\u093e \u090f\u0932\u093f\u092e\u0947\u0902\u091f <span class=\"katex\"><span class=\"katex-mathml\">A\u22121A^{-1}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">A<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u091c\u093f\u0938\u0938\u0947 <span class=\"katex\"><span class=\"katex-mathml\">A\u2217A\u22121=IdentityA * A^{-1} = Identity<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">A<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">I<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span> \u0939\u094b\u0964<\/p>\n<h3 data-start=\"2714\" data-end=\"2746\"><strong data-start=\"2718\" data-end=\"2746\">\u00a0Conclusion (\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937)<\/strong><\/h3>\n<ul data-start=\"2747\" data-end=\"3021\">\n<li data-start=\"2747\" data-end=\"2798\">Binary Operator <strong data-start=\"2765\" data-end=\"2787\">\u0926\u094b \u0935\u0948\u0932\u094d\u092f\u0942 \u092a\u0930 \u0915\u093e\u0930\u094d\u092f<\/strong> \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/li>\n<li data-start=\"2799\" data-end=\"2873\">\u092f\u0939 <strong data-start=\"2804\" data-end=\"2846\">\u0917\u0923\u093f\u0924\u0940\u092f, \u0924\u093e\u0930\u094d\u0915\u093f\u0915, \u0930\u093f\u0932\u0947\u0936\u0928\u0932 \u0914\u0930 \u0938\u0947\u091f \u0911\u092a\u0930\u0947\u0936\u0928<\/strong> \u092e\u0947\u0902 \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/li>\n<li data-start=\"2874\" data-end=\"2944\">\u0907\u0938\u0915\u0940 <strong data-start=\"2881\" data-end=\"2932\">Closure, Associative, \u0914\u0930 Commutative Properties<\/strong> \u0939\u094b\u0924\u0940 \u0939\u0948\u0902\u0964<\/li>\n<li data-start=\"2945\" data-end=\"3021\">\u092f\u0939 <strong data-start=\"2950\" data-end=\"3000\">GATE, Discrete Mathematics \u0914\u0930 Computer Science<\/strong> \u092e\u0947\u0902 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0964<\/li>\n<\/ul>\n<p data-start=\"3023\" data-end=\"3076\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"3026\" data-end=\"3076\" data-is-last-node=\"\">\u0915\u094d\u092f\u093e \u0906\u092a\u0915\u094b \u0914\u0930 \u092a\u094d\u0930\u0936\u094d\u0928 \u091a\u093e\u0939\u093f\u090f GATE 2025 \u0915\u0947 \u0932\u093f\u090f?<\/strong><\/p>\n<h3 data-start=\"3023\" data-end=\"3076\"><a href=\"https:\/\/www2.cs.uh.edu\/~arjun\/courses\/ds\/DiscMaths4CompSc.pdf\" target=\"_blank\" rel=\"noopener\">Binary Operator concept- GATE 2025 Discrete mathematics tutorial in Hindi The binary operator<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/mrce.in\/ebooks\/Maths-Discrete%20Mathematics%20&amp;%20its%20Applications%208th%20Ed.pdf\" target=\"_blank\" rel=\"noopener\">Discrete Mathematics and Its Applications, Eighth Edition<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.vidyalankar.org\/gate\/assets\/docs\/notes\/maths.pdf\" target=\"_blank\" rel=\"noopener\">Engineering Mathematics Notes<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/udrc.lkouniv.ac.in\/Content\/DepartmentContent\/SY_ccb5cce0-bd82-4ee7-8c2e-9f26ae0fb01d_58.pdf\" target=\"_blank\" rel=\"noopener\">UNIVERSITY OF LUCKNOW<\/a><\/h3>\n<p data-start=\"0\" data-end=\"105\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 (Binary Operation) \u0921\u093f\u0938\u094d\u0915\u094d\u0930\u0940\u091f \u092e\u0948\u0925\u092e\u0947\u091f\u093f\u0915\u094d\u0938 \u0915\u093e \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0937\u092f \u0939\u0948, \u091c\u094b GATE 2025 \u091c\u0948\u0938\u0940 \u092a\u0930\u0940\u0915\u094d\u0937\u093e\u0913\u0902 \u092e\u0947\u0902 \u0905\u0915\u094d\u0938\u0930 \u092a\u0942\u091b\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span> \u0906\u0907\u090f \u0907\u0938\u0947 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u0938\u092e\u091d\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<hr data-start=\"107\" data-end=\"110\" \/>\n<h2 data-start=\"112\" data-end=\"140\">\ud83d\udd39 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u094d\u092f\u093e \u0939\u0948?<\/h2>\n<p data-start=\"142\" data-end=\"216\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u091c\u092c \u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0915\u0947 \u0926\u094b \u090f\u0932\u093f\u092e\u0947\u0902\u091f\u094d\u0938 \u0915\u094b \u090f\u0915 \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092e\u093f\u0932\u093e\u0915\u0930 \u0909\u0938\u0940 \u0938\u0947\u091f \u0915\u093e \u090f\u0915 \u0928\u092f\u093e \u090f\u0932\u093f\u092e\u0947\u0902\u091f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948, \u0924\u094b \u0909\u0938 \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u094b <strong data-start=\"120\" data-end=\"137\">\u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n<p data-start=\"218\" data-end=\"238\"><strong data-start=\"218\" data-end=\"238\">\u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e:<\/strong><\/p>\n<p data-start=\"240\" data-end=\"316\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u094b\u0908 \u0938\u0947\u091f \u0939\u0948, \u0924\u094b \u090f\u0915 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 <span class=\"katex\"><span class=\"katex-mathml\">\u2217*<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2217<\/span><\/span><\/span><\/span> \u0910\u0938\u093e \u092b\u0902\u0915\u094d\u0936\u0928 \u0939\u0948:<\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2217:S\u00d7S\u2192S* : S \\times S \\rightarrow S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2217<\/span><span class=\"mrel\">:<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"354\" data-end=\"432\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0926\u094b \u090f\u0932\u093f\u092e\u0947\u0902\u091f\u094d\u0938 <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> \u0914\u0930 <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> \u0915\u0947 \u0932\u093f\u090f, <span class=\"katex\"><span class=\"katex-mathml\">a\u2217ba * b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> \u092d\u0940 <span class=\"katex\"><span class=\"katex-mathml\">SS<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u092e\u0947\u0902 \u0939\u0940 \u0939\u094b\u0917\u093e\u0964<\/span><\/p>\n<hr data-start=\"434\" data-end=\"437\" \/>\n<h2 data-start=\"439\" data-end=\"451\">\ud83d\udd39 \u0909\u0926\u093e\u0939\u0930\u0923<\/h2>\n<ol data-start=\"453\" data-end=\"842\">\n<li data-start=\"453\" data-end=\"631\">\n<p data-start=\"456\" data-end=\"502\"><strong data-start=\"456\" data-end=\"502\">\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u091c\u094b\u0921\u093c (Addition on Integers):<\/strong><\/p>\n<p data-start=\"507\" data-end=\"631\">\u0938\u0947\u091f <span class=\"katex\"><span class=\"katex-mathml\">Z\\mathbb{Z}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathbb\">Z<\/span><\/span><\/span><\/span> (\u0938\u092d\u0940 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915) \u092a\u0930 \u091c\u094b\u0921\u093c \u090f\u0915 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0915\u093f\u0938\u0940 \u092d\u0940 \u0926\u094b \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u093e \u092f\u094b\u0917 \u092d\u0940 \u090f\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<\/li>\n<li data-start=\"633\" data-end=\"842\">\n<p data-start=\"636\" data-end=\"700\"><strong data-start=\"636\" data-end=\"700\">\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092a\u0930 \u0918\u091f\u093e\u0935 (Subtraction on Natural Numbers):<\/strong><\/p>\n<p data-start=\"705\" data-end=\"842\">\u0938\u0947\u091f <span class=\"katex\"><span class=\"katex-mathml\">N\\mathbb{N}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathbb\">N<\/span><\/span><\/span><\/span> (\u0938\u092d\u0940 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0902) \u092a\u0930 \u0918\u091f\u093e\u0935 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex\"><span class=\"katex-mathml\">3\u22125=\u221223 &#8211; 5 = -2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>, \u091c\u094b \u0915\u093f <span class=\"katex\"><span class=\"katex-mathml\">N\\mathbb{N}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathbb\">N<\/span><\/span><\/span><\/span> \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"844\" data-end=\"847\" \/>\n<h2 data-start=\"849\" data-end=\"875\">\ud83d\udd39 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u0947 \u0917\u0941\u0923<\/h2>\n<ol data-start=\"877\" data-end=\"1441\">\n<li data-start=\"877\" data-end=\"980\">\n<p data-start=\"880\" data-end=\"980\"><strong data-start=\"880\" data-end=\"901\">\u0915\u094d\u0932\u094b\u091c\u0930 (Closure):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">a,b\u2208Sa, b \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span>, \u0924\u094b <span class=\"katex\"><span class=\"katex-mathml\">a\u2217b\u2208Sa * b \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"982\" data-end=\"1094\">\n<p data-start=\"985\" data-end=\"1094\"><strong data-start=\"985\" data-end=\"1015\">\u0938\u0939\u0938\u0902\u092c\u0902\u0927\u0924\u093e (Associativity):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">(a\u2217b)\u2217c=a\u2217(b\u2217c)(a * b) * c = a * (b * c)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> \u0938\u092d\u0940 <span class=\"katex\"><span class=\"katex-mathml\">a,b,c\u2208Sa, b, c \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0938\u0924\u094d\u092f \u0939\u094b, \u0924\u094b \u0911\u092a\u0930\u0947\u0936\u0928 \u090f\u0938\u094b\u0938\u093f\u090f\u091f\u093f\u0935 \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1096\" data-end=\"1206\">\n<p data-start=\"1099\" data-end=\"1206\"><strong data-start=\"1099\" data-end=\"1127\">\u0938\u092e\u092e\u093f\u0924\u0924\u093e (Commutativity):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f <span class=\"katex\"><span class=\"katex-mathml\">a\u2217b=b\u2217aa * b = b * a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> \u0938\u092d\u0940 <span class=\"katex\"><span class=\"katex-mathml\">a,b\u2208Sa, b \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0938\u0924\u094d\u092f \u0939\u094b, \u0924\u094b \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u092e\u094d\u092f\u0942\u091f\u0947\u091f\u093f\u0935 \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1208\" data-end=\"1324\">\n<p data-start=\"1211\" data-end=\"1324\"><strong data-start=\"1211\" data-end=\"1245\">\u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (Identity Element):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f \u0915\u094b\u0908 <span class=\"katex\"><span class=\"katex-mathml\">e\u2208Se \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0910\u0938\u093e \u0939\u094b \u0915\u093f <span class=\"katex\"><span class=\"katex-mathml\">a\u2217e=e\u2217a=aa * e = e * a = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> \u0938\u092d\u0940 <span class=\"katex\"><span class=\"katex-mathml\">a\u2208Sa \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u0932\u093f\u090f, \u0924\u094b <span class=\"katex\"><span class=\"katex-mathml\">ee<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><\/span><\/span><\/span> \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1326\" data-end=\"1441\">\n<p data-start=\"1329\" data-end=\"1441\"><strong data-start=\"1329\" data-end=\"1362\">\u0909\u0932\u094d\u091f\u093e \u0924\u0924\u094d\u0935 (Inverse Element):<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0926\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 <span class=\"katex\"><span class=\"katex-mathml\">a\u2208Sa \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0915\u094b\u0908 <span class=\"katex\"><span class=\"katex-mathml\">b\u2208Sb \\in S<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">\u2208<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">S<\/span><\/span><\/span><\/span> \u0910\u0938\u093e \u0939\u094b \u0915\u093f <span class=\"katex\"><span class=\"katex-mathml\">a\u2217b=b\u2217a=ea * b = b * a = e<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2217<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><\/span><\/span><\/span>, \u091c\u0939\u093e\u0901 <span class=\"katex\"><span class=\"katex-mathml\">ee<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">e<\/span><\/span><\/span><\/span> \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948, \u0924\u094b <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> \u0915\u094b <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> \u0915\u093e \u0909\u0932\u094d\u091f\u093e \u0924\u0924\u094d\u0935 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"1443\" data-end=\"1446\" \/>\n<h2 data-start=\"1448\" data-end=\"1477\">\ud83d\udd39 GATE 2025 \u0915\u0947 \u0932\u093f\u090f \u0924\u0948\u092f\u093e\u0930\u0940<\/h2>\n<p data-start=\"1479\" data-end=\"1557\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">GATE \u092a\u0930\u0940\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 \u0905\u0915\u094d\u0938\u0930 \u092a\u0942\u091b\u0947 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947:<\/span><\/p>\n<ul data-start=\"1559\" data-end=\"1803\">\n<li data-start=\"1559\" data-end=\"1639\">\n<p data-start=\"1561\" data-end=\"1639\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0915\u093f\u0938\u0940 \u0926\u093f\u090f \u0917\u090f \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u0947 \u0932\u093f\u090f \u0915\u094d\u0932\u094b\u091c\u0930, \u090f\u0938\u094b\u0938\u093f\u090f\u091f\u093f\u0935\u093f\u091f\u0940, \u0915\u092e\u094d\u092f\u0942\u091f\u0947\u091f\u093f\u0935\u093f\u091f\u0940 \u0915\u0940 \u091c\u093e\u0902\u091a \u0915\u0930\u0928\u093e\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1641\" data-end=\"1721\">\n<p data-start=\"1643\" data-end=\"1721\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0914\u0930 \u0909\u0932\u094d\u091f\u093e \u0924\u0924\u094d\u0935 \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u0930\u0928\u093e\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1723\" data-end=\"1803\">\n<p data-start=\"1725\" data-end=\"1803\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0915\u093f\u0938\u0940 \u0938\u0947\u091f \u0914\u0930 \u0911\u092a\u0930\u0947\u0936\u0928 \u0915\u0947 \u0932\u093f\u090f \u0917\u094d\u0930\u0941\u092a, \u0938\u0947\u092e\u0940-\u0917\u094d\u0930\u0941\u092a, \u092e\u094b\u0928\u0949\u0907\u0921 \u0906\u0926\u093f \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u0930\u0928\u093e\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1805\" data-end=\"1808\" \/>\n<h2 data-start=\"1810\" data-end=\"1831\">\ud83d\udcda \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924 \u0938\u0902\u0938\u093e\u0927\u0928<\/h2>\n<p data-start=\"1833\" data-end=\"1911\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092c\u093e\u0907\u0928\u0930\u0940 \u0911\u092a\u0930\u0947\u0936\u0928 \u0914\u0930 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0935\u093f\u0937\u092f\u094b\u0902 \u0915\u094b \u0914\u0930 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0939\u093f\u0902\u0926\u0940 \u091f\u094d\u092f\u0942\u091f\u094b\u0930\u093f\u092f\u0932\u094d\u0938 \u0938\u0939\u093e\u092f\u0915 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/span><\/p>\n<ul data-start=\"1913\" data-end=\"2243\">\n<li data-start=\"1913\" data-end=\"2020\">\n<p data-start=\"1915\" data-end=\"2020\"><strong data-start=\"1915\" data-end=\"1980\">Binary Operation Group Theory | Discrete Mathematics | Hindi:<\/strong> <span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">YouTube<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"2022\" data-end=\"2128\">\n<p data-start=\"2024\" data-end=\"2128\"><strong data-start=\"2024\" data-end=\"2088\">Algebraic Structure, Binary Operations and Closure Property:<\/strong> <span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">YouTube<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"2130\" data-end=\"2243\">\n<p data-start=\"2132\" data-end=\"2243\"><strong data-start=\"2132\" data-end=\"2203\">Binary Operation, n-ary Operation &amp; Algebraic Structure by Dr.D.N.:<\/strong> <span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">YouTube<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2245\" data-end=\"2248\" \/>\n<p data-start=\"2250\" data-end=\"2325\">\u092f\u0926\u093f \u0906\u092a \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 \u092f\u093e \u0909\u0926\u093e\u0939\u0930\u0923 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0915\u0943\u092a\u092f\u093e \u092c\u0924\u093e\u090f\u0902!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Binary Operator concept- GATE 2021 Discrete mathematics tutorial in Hindi The binary operator [fvplayer id=&#8221;178&#8243;] \u00a0Binary Operator Concept &#8211; GATE 2025 Discrete Mathematics Tutorial in Hindi \u00a0\u0915\u094d\u092f\u093e \u0939\u0948 Binary Operator? Binary Operator \u090f\u0915 \u0910\u0938\u093e \u0911\u092a\u0930\u0947\u091f\u0930 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u0926\u094b \u0911\u092a\u0930\u0947\u091f\u093f\u0902\u0917 \u0935\u0948\u0932\u094d\u092f\u0942 (Operands) \u092a\u0930 \u0915\u093e\u092e \u0915\u0930\u0924\u093e \u0939\u0948 \u0914\u0930 \u090f\u0915 \u0938\u093f\u0902\u0917\u0932 \u0930\u093f\u091c\u0932\u094d\u091f \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0947 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2936","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2936","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2936"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2936\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2936"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2936"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2936"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}