{"id":2943,"date":"2025-06-05T07:07:49","date_gmt":"2025-06-05T07:07:49","guid":{"rendered":"https:\/\/diznr.com\/?p=2943"},"modified":"2025-06-05T07:07:49","modified_gmt":"2025-06-05T07:07:49","slug":"tautology-concept-gate-2021-previous-year-paper-discrete-mathematics-consider-following-the","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/tautology-concept-gate-2021-previous-year-paper-discrete-mathematics-consider-following-the\/","title":{"rendered":"Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following"},"content":{"rendered":"<p>Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following<\/p>\n<p>[fvplayer id=&#8221;182&#8243;]<\/p>\n<p data-start=\"0\" data-end=\"422\">A <strong data-start=\"2\" data-end=\"15\">tautology<\/strong> in propositional logic is a formula that is true in every possible interpretation, meaning it always evaluates to true regardless of the truth values of its constituent propositions. For example, the statement <span class=\"katex\"><span class=\"katex-mathml\">P\u2228\u00acPP \\vee \\neg P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> (where <span class=\"katex\"><span class=\"katex-mathml\">\u2228\\vee<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2228<\/span><\/span><\/span><\/span> denotes logical OR and <span class=\"katex\"><span class=\"katex-mathml\">\u00ac\\neg<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><\/span><\/span><\/span> denotes logical NOT) is a tautology because either <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> is true or <span class=\"katex\"><span class=\"katex-mathml\">\u00acP\\neg P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> is true, covering all possible scenarios.<\/p>\n<p data-start=\"424\" data-end=\"663\">In the context of the GATE 2025 examination, questions related to tautologies often require determining whether a given propositional formula is always true. For instance, consider the following question from the GATE CSE 2025 Set 1 paper:<\/p>\n<p data-start=\"665\" data-end=\"730\"><em data-start=\"665\" data-end=\"730\">Which one of the following well-formed formulae is a tautology?<\/em><\/p>\n<ol data-start=\"732\" data-end=\"941\">\n<li data-start=\"732\" data-end=\"798\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2192(\u00acQ\u2192\u00acP)(P \\rightarrow Q) \\rightarrow (\\neg Q \\rightarrow \\neg P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"799\" data-end=\"855\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2192(Q\u2192P)(P \\rightarrow Q) \\rightarrow (Q \\rightarrow P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"856\" data-end=\"896\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2192(Q\u2192P)P \\rightarrow (Q \\rightarrow P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"897\" data-end=\"941\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2227Q)\u2192(P\u2228Q)(P \\wedge Q) \\rightarrow (P \\vee Q)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<p data-start=\"943\" data-end=\"956\"><strong data-start=\"943\" data-end=\"956\">Solution:<\/strong><\/p>\n<ul data-start=\"958\" data-end=\"1829\">\n<li data-start=\"958\" data-end=\"1198\">\n<p data-start=\"960\" data-end=\"1033\">Option 1: <span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2192(\u00acQ\u2192\u00acP)(P \\rightarrow Q) \\rightarrow (\\neg Q \\rightarrow \\neg P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"1036\" data-end=\"1198\">\n<li data-start=\"1036\" data-end=\"1198\">This is the contrapositive law, which states that a conditional statement is logically equivalent to its contrapositive. Therefore, this formula is a tautology.<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1200\" data-end=\"1415\">\n<p data-start=\"1202\" data-end=\"1265\">Option 2: <span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2192(Q\u2192P)(P \\rightarrow Q) \\rightarrow (Q \\rightarrow P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"1268\" data-end=\"1415\">\n<li data-start=\"1268\" data-end=\"1415\">This suggests that if <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> implies <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span> implies <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span>. This is not necessarily true in all cases, so it is not a tautology.<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1417\" data-end=\"1639\">\n<p data-start=\"1419\" data-end=\"1466\">Option 3: <span class=\"katex\"><span class=\"katex-mathml\">P\u2192(Q\u2192P)P \\rightarrow (Q \\rightarrow P)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"1469\" data-end=\"1639\">\n<li data-start=\"1469\" data-end=\"1639\">This indicates that if <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> is true, then <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span> implies <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span>. Regardless of <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span>&#8216;s value, <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> remains true if <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> is true. Thus, this is a tautology.<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1641\" data-end=\"1829\">\n<p data-start=\"1643\" data-end=\"1694\">Option 4: <span class=\"katex\"><span class=\"katex-mathml\">(P\u2227Q)\u2192(P\u2228Q)(P \\wedge Q) \\rightarrow (P \\vee Q)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"1697\" data-end=\"1829\">\n<li data-start=\"1697\" data-end=\"1829\">If both <span class=\"katex\"><span class=\"katex-mathml\">PP<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">QQ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span> are true, then at least one of them is true. This is always the case, making this formula a tautology.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p data-start=\"1831\" data-end=\"2294\">Therefore, options 1, 3, and 4 are tautologies. However, since the question asks for the one well-formed formula that is a tautology, and multiple options satisfy this condition, it might be a misinterpretation or misprint in the question. Typically, such questions are designed to have a single correct answer. Candidates should carefully analyze each option and refer to authoritative sources or errata provided by the examination authorities for clarification.<\/p>\n<p data-start=\"2296\" data-end=\"2630\">For more practice on such topics, you can refer to previous years&#8217; GATE questions on mathematical logic and tautologies. Websites like ExamSIDE provide a collection of past GATE questions with solutions, which can be beneficial for understanding the pattern and difficulty level of the questions.<\/p>\n<h3 data-start=\"2296\" data-end=\"2630\"><a href=\"https:\/\/www2.cs.uh.edu\/~arjun\/courses\/ds\/DiscMaths4CompSc.pdf\" target=\"_blank\" rel=\"noopener\">Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following<\/a><\/h3>\n<p data-start=\"0\" data-end=\"74\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">GATE 2025 \u0915\u0940 \u0924\u0948\u092f\u093e\u0930\u0940 \u0915\u0930 \u0930\u0939\u0947 \u091b\u093e\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f, <strong data-start=\"43\" data-end=\"56\">Tautology<\/strong> (\u0938\u0924\u094d\u092f\u0935\u093e\u0915\u094d\u092f) \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0937\u092f \u0939\u0948, \u091c\u094b \u0905\u0915\u094d\u0938\u0930 <strong data-start=\"101\" data-end=\"125\">Discrete Mathematics<\/strong> \u0915\u0947 \u0905\u0902\u0924\u0930\u094d\u0917\u0924 \u092a\u0942\u091b\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n<hr data-start=\"76\" data-end=\"79\" \/>\n<h2 data-start=\"81\" data-end=\"109\">\ud83d\udcd8 <strong data-start=\"87\" data-end=\"109\">Tautology \u0915\u094d\u092f\u093e \u0939\u0948?<\/strong><\/h2>\n<p data-start=\"111\" data-end=\"185\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"13\" data-is-only-node=\"\">Tautology<\/strong> \u090f\u0915 \u0910\u0938\u093e \u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0915\u0925\u0928 (logical statement) \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u0905\u092a\u0928\u0947 \u0938\u092d\u0940 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0938\u0924\u094d\u092f \u092e\u093e\u0928\u094b\u0902 (truth values) \u0915\u0947 \u0932\u093f\u090f \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f (true) \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n<p data-start=\"187\" data-end=\"200\"><strong data-start=\"187\" data-end=\"198\">\u0909\u0926\u093e\u0939\u0930\u0923:<\/strong><\/p>\n<ul data-start=\"201\" data-end=\"279\">\n<li data-start=\"201\" data-end=\"279\">\n<p data-start=\"203\" data-end=\"279\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"10\" data-is-only-node=\"\">p \u2228 \u00acp<\/strong> (\u092f\u0939 \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f \u0939\u094b\u0924\u093e \u0939\u0948, \u091a\u093e\u0939\u0947 p \u0915\u093e \u092e\u093e\u0928 \u0915\u0941\u091b \u092d\u0940 \u0939\u094b)<\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"281\" data-end=\"284\" \/>\n<h2 data-start=\"286\" data-end=\"340\">\ud83d\udcda <strong data-start=\"292\" data-end=\"340\">GATE \u092e\u0947\u0902 \u092a\u0942\u091b\u0947 \u0917\u090f Tautology \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928<\/strong><\/h2>\n<p data-start=\"342\" data-end=\"420\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">GATE CSE 2002 \u092e\u0947\u0902 \u090f\u0915 \u092a\u094d\u0930\u0936\u094d\u0928 \u092a\u0942\u091b\u093e \u0917\u092f\u093e \u0925\u093e:<\/span><\/p>\n<p data-start=\"422\" data-end=\"512\"><strong data-start=\"422\" data-end=\"433\">\u092a\u094d\u0930\u0936\u094d\u0928:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092e\u0947\u0902 \u0938\u0947 \u0915\u094c\u0928-\u0938\u093e tautology \u0939\u0948?<\/span><\/p>\n<ol data-start=\"514\" data-end=\"685\">\n<li data-start=\"514\" data-end=\"558\">\n<p data-start=\"517\" data-end=\"558\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"15\" data-is-last-node=\"\" data-is-only-node=\"\">A \u2194 (A \u2228 A)<\/strong><\/span><\/p>\n<\/li>\n<li data-start=\"559\" data-end=\"603\">\n<p data-start=\"562\" data-end=\"603\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"15\" data-is-last-node=\"\" data-is-only-node=\"\">(A \u2228 B) \u2192 B<\/strong><\/span><\/p>\n<\/li>\n<li data-start=\"604\" data-end=\"685\">\n<p data-start=\"607\" data-end=\"685\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"16\" data-is-last-node=\"\" data-is-only-node=\"\">A \u2228 \u00ac(A \u2228 B)<\/strong><\/span><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<\/ol>\n<p data-start=\"687\" data-end=\"697\"><strong data-start=\"687\" data-end=\"697\">\u0909\u0924\u094d\u0924\u0930:<\/strong><\/p>\n<ol data-start=\"699\" data-end=\"1160\">\n<li data-start=\"699\" data-end=\"851\">\n<p data-start=\"702\" data-end=\"720\"><strong data-start=\"702\" data-end=\"718\">A \u2194 (A \u2228 A):<\/strong><\/p>\n<ul data-start=\"724\" data-end=\"851\">\n<li data-start=\"724\" data-end=\"767\">\n<p data-start=\"726\" data-end=\"767\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A \u2228 A = A (Idempotent Law)<\/span><\/p>\n<\/li>\n<li data-start=\"771\" data-end=\"851\">\n<p data-start=\"773\" data-end=\"851\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0938\u0932\u093f\u090f, A \u2194 A = <strong data-start=\"15\" data-end=\"28\" data-is-last-node=\"\">Tautology<\/strong><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"853\" data-end=\"1005\">\n<p data-start=\"856\" data-end=\"874\"><strong data-start=\"856\" data-end=\"872\">(A \u2228 B) \u2192 B:<\/strong><\/p>\n<ul data-start=\"878\" data-end=\"1005\">\n<li data-start=\"878\" data-end=\"921\">\n<p data-start=\"880\" data-end=\"921\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0939 \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e; \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0926\u093f A = True \u0914\u0930 B = False, \u0924\u094b A \u2228 B = True, \u0932\u0947\u0915\u093f\u0928 B = False, \u0907\u0938\u0932\u093f\u090f \u092a\u0942\u0930\u093e \u0915\u0925\u0928 False \u0939\u094b\u0917\u093e\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"925\" data-end=\"1005\">\n<p data-start=\"927\" data-end=\"1005\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0938\u0932\u093f\u090f, \u092f\u0939 <strong data-start=\"10\" data-end=\"28\">Tautology \u0928\u0939\u0940\u0902<\/strong> \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1007\" data-end=\"1160\">\n<p data-start=\"1010\" data-end=\"1029\"><strong data-start=\"1010\" data-end=\"1027\">A \u2228 \u00ac(A \u2228 B):<\/strong><\/p>\n<ul data-start=\"1033\" data-end=\"1160\">\n<li data-start=\"1033\" data-end=\"1076\">\n<p data-start=\"1035\" data-end=\"1076\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u092f\u0939 \u0915\u0925\u0928 \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e; \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0926\u093f A = False \u0914\u0930 B = True, \u0924\u094b A \u2228 B = True, \u00ac(A \u2228 B) = False, \u0914\u0930 A \u2228 False = False\u0964<\/span><\/p>\n<\/li>\n<li data-start=\"1080\" data-end=\"1160\">\n<p data-start=\"1082\" data-end=\"1160\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0907\u0938\u0932\u093f\u090f, \u092f\u0939 \u092d\u0940 <strong data-start=\"13\" data-end=\"31\">Tautology \u0928\u0939\u0940\u0902<\/strong> \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"1162\" data-end=\"1212\"><strong data-start=\"1162\" data-end=\"1172\">\u0938\u094d\u0930\u094b\u0924:<\/strong><\/p>\n<hr data-start=\"1214\" data-end=\"1217\" \/>\n<h2 data-start=\"1219\" data-end=\"1258\">\ud83e\udde0 <strong data-start=\"1225\" data-end=\"1258\">Tautology \u0915\u0940 \u092a\u0939\u091a\u093e\u0928 \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902?<\/strong><\/h2>\n<ol data-start=\"1260\" data-end=\"1542\">\n<li data-start=\"1260\" data-end=\"1392\">\n<p data-start=\"1263\" data-end=\"1308\"><strong data-start=\"1263\" data-end=\"1306\">\u0938\u0924\u094d\u092f \u0938\u093e\u0930\u0923\u0940 (Truth Table) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902:<\/strong><\/p>\n<ul data-start=\"1312\" data-end=\"1392\">\n<li data-start=\"1312\" data-end=\"1392\">\n<p data-start=\"1314\" data-end=\"1392\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0938\u092d\u0940 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u092e\u093e\u0928\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0915\u0925\u0928 \u0915\u093e \u092e\u0942\u0932\u094d\u092f\u093e\u0902\u0915\u0928 \u0915\u0930\u0947\u0902\u0964 \u092f\u0926\u093f \u0938\u092d\u0940 \u092e\u093e\u092e\u0932\u094b\u0902 \u092e\u0947\u0902 \u092a\u0930\u093f\u0923\u093e\u092e True \u0939\u0948, \u0924\u094b \u0935\u0939 Tautology \u0939\u0948\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1394\" data-end=\"1542\">\n<p data-start=\"1397\" data-end=\"1458\"><strong data-start=\"1397\" data-end=\"1456\">\u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0938\u092e\u0924\u0941\u0932\u094d\u092f\u0924\u093e (Logical Equivalences) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902:<\/strong><\/p>\n<ul data-start=\"1462\" data-end=\"1542\">\n<li data-start=\"1462\" data-end=\"1542\">\n<p data-start=\"1464\" data-end=\"1542\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0915\u0941\u091b \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0915\u0925\u0928 \u0915\u094b \u0938\u0930\u0932 \u092c\u0928\u093e\u090f\u0902\u0964<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"1544\" data-end=\"1557\"><strong data-start=\"1544\" data-end=\"1555\">\u0909\u0926\u093e\u0939\u0930\u0923:<\/strong><\/p>\n<ul data-start=\"1558\" data-end=\"1682\">\n<li data-start=\"1558\" data-end=\"1601\">\n<p data-start=\"1560\" data-end=\"1601\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"22\" data-is-only-node=\"\">p \u2228 \u00acp = Tautology<\/strong> (Law of the Excluded Middle)<\/span><\/p>\n<\/li>\n<li data-start=\"1602\" data-end=\"1682\">\n<p data-start=\"1604\" data-end=\"1682\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"26\" data-is-only-node=\"\">p \u2227 \u00acp = Contradiction<\/strong> (Law of Non-Contradiction)<\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1684\" data-end=\"1687\" \/>\n<h2 data-start=\"1689\" data-end=\"1728\">\ud83c\udfa5 <strong data-start=\"1695\" data-end=\"1728\">\u0905\u0927\u093f\u0915 \u0938\u092e\u091d \u0915\u0947 \u0932\u093f\u090f \u0935\u0940\u0921\u093f\u092f\u094b \u0938\u0902\u0938\u093e\u0927\u0928<\/strong><\/h2>\n<p data-start=\"1730\" data-end=\"1808\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">\u0906\u092a \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0935\u0940\u0921\u093f\u092f\u094b \u092e\u0947\u0902 GATE \u0915\u0947 \u092a\u093f\u091b\u0932\u0947 \u0935\u0930\u094d\u0937\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u0936\u094d\u0928\u094b\u0902 \u0915\u093e \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0926\u0947\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/span><\/p>\n<div class=\"not-prose mb-3 flex flex-col gap-4 text-base\">\n<div><\/div>\n<\/div>\n<hr data-start=\"1854\" data-end=\"1857\" \/>\n<p data-start=\"1859\" data-end=\"1974\">\u092f\u0926\u093f \u0906\u092a Tautology \u092a\u0930 \u0914\u0930 \u0905\u0927\u093f\u0915 \u0905\u092d\u094d\u092f\u093e\u0938 \u092a\u094d\u0930\u0936\u094d\u0928 \u092f\u093e \u0938\u094d\u092a\u0937\u094d\u091f\u0940\u0915\u0930\u0923 \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0915\u0943\u092a\u092f\u093e \u092c\u0924\u093e\u090f\u0902\u0964 \u092e\u0948\u0902 \u0906\u092a\u0915\u0940 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0947 \u0932\u093f\u090f \u092f\u0939\u093e\u0901 \u0939\u0942\u0901!<\/p>\n<h3 data-start=\"1859\" data-end=\"1974\"><a href=\"https:\/\/www.aicte-india.org\/sites\/default\/files\/MQP.pdf\" target=\"_blank\" rel=\"noopener\">Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/sriindu.ac.in\/wp-content\/uploads\/2023\/10\/R20CSE2201-DISCRETE-MATHEMATICS.pdf\" target=\"_blank\" rel=\"noopener\">DISCRETE MATHEMATICS<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Tautology Concept- GATE 2025- Previous year paper discrete mathematics consider the following [fvplayer id=&#8221;182&#8243;] A tautology in propositional logic is a formula that is true in every possible interpretation, meaning it always evaluates to true regardless of the truth values of its constituent propositions. For example, the statement P\u2228\u00acPP \\vee \\neg PP\u2228\u00acP (where \u2228\\vee\u2228 denotes [&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-2943","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2943","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=2943"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2943\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2943"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2943"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2943"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}