{"id":2983,"date":"2025-06-01T15:08:58","date_gmt":"2025-06-01T15:08:58","guid":{"rendered":"https:\/\/diznr.com\/?p=2983"},"modified":"2025-06-01T15:08:58","modified_gmt":"2025-06-01T15:08:58","slug":"day-04part12-concept-of-tautology-contradiction-and-contigency-satisfiable-and-unsatisfiable-case","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-04part12-concept-of-tautology-contradiction-and-contigency-satisfiable-and-unsatisfiable-case\/","title":{"rendered":"Day 04Part12-Concept of tautology contradiction and Contigency , Satisfiable and Unsatisfiable case."},"content":{"rendered":"<p>Day 04Part12-Concept of tautology contradiction and Contigency, Satisfiable and Unsatisfiable case.<\/p>\n<p>[fvplayer id=&#8221;201&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"79\"><strong data-start=\"3\" data-end=\"77\">Day 04 &#8211; Part 12: Concept of Tautology, Contradiction, and Contingency<\/strong><\/h3>\n<p data-start=\"81\" data-end=\"326\">In <strong data-start=\"84\" data-end=\"118\">Discrete Mathematics and Logic<\/strong>, statements can be categorized based on their truth values in all possible scenarios. The key concepts are <strong data-start=\"226\" data-end=\"271\">Tautology, Contradiction, and Contingency<\/strong>, along with <strong data-start=\"284\" data-end=\"323\">Satisfiable and Unsatisfiable cases<\/strong>.<\/p>\n<h3 data-start=\"333\" data-end=\"383\"><strong data-start=\"337\" data-end=\"381\">1. Tautology (\u0938\u0930\u094d\u0935\u0926\u093e \u0938\u0924\u094d\u092f \/ Always True)<\/strong><\/h3>\n<p data-start=\"384\" data-end=\"510\">A <strong data-start=\"386\" data-end=\"399\">tautology<\/strong> is a <strong data-start=\"405\" data-end=\"446\">logical statement that is always true<\/strong>, regardless of the truth values of its individual components.<\/p>\n<p data-start=\"512\" data-end=\"529\"><strong data-start=\"515\" data-end=\"527\">Example:<\/strong><\/p>\n<ul data-start=\"530\" data-end=\"631\">\n<li data-start=\"530\" data-end=\"578\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2228\u00acpp \\lor \\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> (Law of Excluded Middle)<\/li>\n<li data-start=\"579\" data-end=\"631\">&#8220;It will either rain or not rain.&#8221; (Always True)<\/li>\n<\/ul>\n<p data-start=\"633\" data-end=\"654\"><strong data-start=\"636\" data-end=\"652\">Truth Table:<\/strong><\/p>\n<table data-start=\"656\" data-end=\"762\">\n<thead data-start=\"656\" data-end=\"706\">\n<tr data-start=\"656\" data-end=\"706\">\n<th data-start=\"656\" data-end=\"666\"><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><\/th>\n<th data-start=\"666\" data-end=\"681\"><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><\/th>\n<th data-start=\"681\" data-end=\"706\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2228\u00acpp \\lor \\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><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"723\" data-end=\"762\">\n<tr data-start=\"723\" data-end=\"742\">\n<td>T<\/td>\n<td>F<\/td>\n<td><strong data-start=\"733\" data-end=\"738\">T<\/strong><\/td>\n<\/tr>\n<tr data-start=\"743\" data-end=\"762\">\n<td>F<\/td>\n<td>T<\/td>\n<td><strong data-start=\"753\" data-end=\"758\">T<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-start=\"764\" data-end=\"844\">Since the last column contains <strong data-start=\"795\" data-end=\"811\">all True (T)<\/strong> values, it is a <strong data-start=\"828\" data-end=\"841\">Tautology<\/strong>.<\/p>\n<h3 data-start=\"851\" data-end=\"907\"><strong data-start=\"855\" data-end=\"905\">2. Contradiction (\u0938\u0930\u094d\u0935\u0926\u093e \u0905\u0938\u0924\u094d\u092f \/ Always False)<\/strong><\/h3>\n<p data-start=\"908\" data-end=\"1028\">A <strong data-start=\"910\" data-end=\"927\">contradiction<\/strong> is a <strong data-start=\"933\" data-end=\"975\">logical statement that is always false<\/strong>, regardless of the truth values of its components.<\/p>\n<p data-start=\"1030\" data-end=\"1047\"><strong data-start=\"1033\" data-end=\"1045\">Example:<\/strong><\/p>\n<ul data-start=\"1048\" data-end=\"1162\">\n<li data-start=\"1048\" data-end=\"1093\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2227\u00acpp \\land \\neg p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> (Self-Contradiction)<\/li>\n<li data-start=\"1094\" data-end=\"1162\">&#8220;It is raining and not raining at the same time.&#8221; (Always False)<\/li>\n<\/ul>\n<p data-start=\"1164\" data-end=\"1185\"><strong data-start=\"1167\" data-end=\"1183\">Truth Table:<\/strong><\/p>\n<table data-start=\"1187\" data-end=\"1294\">\n<thead data-start=\"1187\" data-end=\"1238\">\n<tr data-start=\"1187\" data-end=\"1238\">\n<th data-start=\"1187\" data-end=\"1197\"><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><\/th>\n<th data-start=\"1197\" data-end=\"1212\"><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><\/th>\n<th data-start=\"1212\" data-end=\"1238\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2227\u00acpp \\land \\neg p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1255\" data-end=\"1294\">\n<tr data-start=\"1255\" data-end=\"1274\">\n<td>T<\/td>\n<td>F<\/td>\n<td><strong data-start=\"1265\" data-end=\"1270\">F<\/strong><\/td>\n<\/tr>\n<tr data-start=\"1275\" data-end=\"1294\">\n<td>F<\/td>\n<td>T<\/td>\n<td><strong data-start=\"1285\" data-end=\"1290\">F<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-start=\"1296\" data-end=\"1381\">Since the last column contains <strong data-start=\"1327\" data-end=\"1344\">all False (F)<\/strong> values, it is a <strong data-start=\"1361\" data-end=\"1378\">Contradiction<\/strong>.<\/p>\n<h3 data-start=\"1388\" data-end=\"1468\"><strong data-start=\"1392\" data-end=\"1466\">3. Contingency (\u0915\u092d\u0940 \u0938\u0924\u094d\u092f, \u0915\u092d\u0940 \u0905\u0938\u0924\u094d\u092f \/ Sometimes True, Sometimes False)<\/strong><\/h3>\n<p data-start=\"1469\" data-end=\"1602\">A <strong data-start=\"1471\" data-end=\"1486\">contingency<\/strong> is a logical statement that is <strong data-start=\"1518\" data-end=\"1556\">sometimes true and sometimes false<\/strong>, depending on the values of its components.<\/p>\n<p data-start=\"1604\" data-end=\"1621\"><strong data-start=\"1607\" data-end=\"1619\">Example:<\/strong><\/p>\n<ul data-start=\"1622\" data-end=\"1676\">\n<li data-start=\"1622\" data-end=\"1676\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2228qp \\lor q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span> (Logical OR between two statements)<\/li>\n<\/ul>\n<p data-start=\"1678\" data-end=\"1699\"><strong data-start=\"1681\" data-end=\"1697\">Truth Table:<\/strong><\/p>\n<table data-start=\"1701\" data-end=\"1837\">\n<thead data-start=\"1701\" data-end=\"1741\">\n<tr data-start=\"1701\" data-end=\"1741\">\n<th data-start=\"1701\" data-end=\"1711\"><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><\/th>\n<th data-start=\"1711\" data-end=\"1721\"><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><\/th>\n<th data-start=\"1721\" data-end=\"1741\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2228qp \\lor q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1758\" data-end=\"1837\">\n<tr data-start=\"1758\" data-end=\"1777\">\n<td>T<\/td>\n<td>T<\/td>\n<td><strong data-start=\"1768\" data-end=\"1773\">T<\/strong><\/td>\n<\/tr>\n<tr data-start=\"1778\" data-end=\"1797\">\n<td>T<\/td>\n<td>F<\/td>\n<td><strong data-start=\"1788\" data-end=\"1793\">T<\/strong><\/td>\n<\/tr>\n<tr data-start=\"1798\" data-end=\"1817\">\n<td>F<\/td>\n<td>T<\/td>\n<td><strong data-start=\"1808\" data-end=\"1813\">T<\/strong><\/td>\n<\/tr>\n<tr data-start=\"1818\" data-end=\"1837\">\n<td>F<\/td>\n<td>F<\/td>\n<td><strong data-start=\"1828\" data-end=\"1833\">F<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-start=\"1839\" data-end=\"1931\">Since the last column has both <strong data-start=\"1870\" data-end=\"1896\">True (T) and False (F)<\/strong> values, it is a <strong data-start=\"1913\" data-end=\"1928\">Contingency<\/strong>.<\/p>\n<h3 data-start=\"1938\" data-end=\"1986\"><strong data-start=\"1942\" data-end=\"1984\">4. Satisfiable and Unsatisfiable Cases<\/strong><\/h3>\n<p data-start=\"1988\" data-end=\"2018\"><strong data-start=\"1990\" data-end=\"2016\">Satisfiable Statement:<\/strong><\/p>\n<ul data-start=\"2019\" data-end=\"2166\">\n<li data-start=\"2019\" data-end=\"2091\">A <strong data-start=\"2023\" data-end=\"2088\">proposition is satisfiable if it is true in at least one case<\/strong>.<\/li>\n<li data-start=\"2092\" data-end=\"2166\">Example: <span class=\"katex\"><span class=\"katex-mathml\">p\u2228qp \\lor q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span> (It has at least one True in the truth table).<\/li>\n<\/ul>\n<p data-start=\"2168\" data-end=\"2200\"><strong data-start=\"2170\" data-end=\"2198\">Unsatisfiable Statement:<\/strong><\/p>\n<ul data-start=\"2201\" data-end=\"2346\">\n<li data-start=\"2201\" data-end=\"2296\">A <strong data-start=\"2205\" data-end=\"2293\">proposition is unsatisfiable if it is never true (always false, i.e., contradiction)<\/strong>.<\/li>\n<li data-start=\"2297\" data-end=\"2346\">Example: <span class=\"katex\"><span class=\"katex-mathml\">p\u2227\u00acpp \\land \\neg p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> (Always False).<\/li>\n<\/ul>\n<h3 data-start=\"2353\" data-end=\"2370\"><strong data-start=\"2357\" data-end=\"2368\">Summary<\/strong><\/h3>\n<table data-start=\"2371\" data-end=\"2797\">\n<thead data-start=\"2371\" data-end=\"2419\">\n<tr data-start=\"2371\" data-end=\"2419\">\n<th data-start=\"2371\" data-end=\"2385\"><strong data-start=\"2373\" data-end=\"2384\">Concept<\/strong><\/th>\n<th data-start=\"2385\" data-end=\"2402\"><strong data-start=\"2387\" data-end=\"2401\">Definition<\/strong><\/th>\n<th data-start=\"2402\" data-end=\"2419\"><strong data-start=\"2404\" data-end=\"2415\">Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2465\" data-end=\"2797\">\n<tr data-start=\"2465\" data-end=\"2520\">\n<td><strong data-start=\"2467\" data-end=\"2480\">Tautology<\/strong><\/td>\n<td>Always True<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">p\u2228\u00acpp \\lor \\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><\/td>\n<\/tr>\n<tr data-start=\"2521\" data-end=\"2582\">\n<td><strong data-start=\"2523\" data-end=\"2540\">Contradiction<\/strong><\/td>\n<td>Always False<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">p\u2227\u00acpp \\land \\neg p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2583\" data-end=\"2655\">\n<td><strong data-start=\"2585\" data-end=\"2600\">Contingency<\/strong><\/td>\n<td>Sometimes True, Sometimes False<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">p\u2228qp \\lor q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2656\" data-end=\"2722\">\n<td><strong data-start=\"2658\" data-end=\"2673\">Satisfiable<\/strong><\/td>\n<td>True in at least one case<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">p\u2228qp \\lor q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2723\" data-end=\"2797\">\n<td><strong data-start=\"2725\" data-end=\"2742\">Unsatisfiable<\/strong><\/td>\n<td>Never True (Always False)<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">p\u2227\u00acpp \\land \\neg p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-start=\"2799\" data-end=\"2879\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like <strong data-start=\"2814\" data-end=\"2875\">more examples or practical applications of these concepts<\/strong>?<\/p>\n<h3 data-start=\"2799\" data-end=\"2879\"><a href=\"https:\/\/www.math.fsu.edu\/~pkirby\/mad2104\/SlideShow\/s2_2.pdf\" target=\"_blank\" rel=\"noopener\">Day 04Part12-Concept of tautology contradiction and Contigency , Satisfiable and Unsatisfiable case.<\/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:\/\/uomustansiriyah.edu.iq\/media\/lectures\/6\/6_2018_02_06!10_11_33_PM.pdf\" target=\"_blank\" rel=\"noopener\">1.3. Tautology \/Contradiction \/ Contingency<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/math.miami.edu\/~wachs\/mth309\/sec_1.3.pdf\" target=\"_blank\" rel=\"noopener\">1.3 Propositional Equivalences<\/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<p>Here is a clear and complete explanation of:<\/p>\n<hr \/>\n<h1>\ud83e\udde0 <strong>Day 04 Part-12: Concepts of Tautology, Contradiction, Contingency, Satisfiable &amp; Unsatisfiable Statements<\/strong><\/h1>\n<p><em>(Discrete Mathematics \/ Logic \u2013 for GATE, BCA, B.Tech, CS)<\/em><\/p>\n<hr \/>\n<h2>\ud83d\udd39 <strong>1. Propositional Logic Recap<\/strong><\/h2>\n<p>A <strong>proposition<\/strong> is a statement that is either <strong>True (T)<\/strong> or <strong>False (F)<\/strong>.<\/p>\n<p>Examples:<\/p>\n<ul>\n<li>&#8220;It is raining.&#8221; \u2705 (Proposition)<\/li>\n<li>&#8220;x + 3 = 7&#8221; \u274c (Not a proposition until x is known)<\/li>\n<\/ul>\n<p>We use logical connectives like:<\/p>\n<ul>\n<li>\u2227 (AND), \u2228 (OR), \u2192 (IMPLIES), \u00ac (NOT)<\/li>\n<\/ul>\n<hr \/>\n<h2>\ud83d\udd39 <strong>2. Tautology (Always True)<\/strong><\/h2>\n<blockquote><p>A <strong>tautology<\/strong> is a <strong>compound statement<\/strong> that is <strong>always true<\/strong> for <strong>all possible truth values<\/strong> of its variables.<\/p><\/blockquote>\n<h3>\u2705 Example:<\/h3>\n<blockquote><p>(P \u2228 \u00acP) is always true.<\/p><\/blockquote>\n<table>\n<thead>\n<tr>\n<th>P<\/th>\n<th>\u00acP<\/th>\n<th>P \u2228 \u00acP<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2714\ufe0f Output is always <strong>True<\/strong> \u2192 It\u2019s a <strong>tautology<\/strong><\/p>\n<hr \/>\n<h2>\ud83d\udd39 <strong>3. Contradiction (Always False)<\/strong><\/h2>\n<blockquote><p>A <strong>contradiction<\/strong> is a <strong>compound statement<\/strong> that is <strong>always false<\/strong>, no matter what the input values are.<\/p><\/blockquote>\n<h3>\u274c Example:<\/h3>\n<blockquote><p>(P \u2227 \u00acP) is always false.<\/p><\/blockquote>\n<table>\n<thead>\n<tr>\n<th>P<\/th>\n<th>\u00acP<\/th>\n<th>P \u2227 \u00acP<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2714\ufe0f Output is always <strong>False<\/strong> \u2192 It\u2019s a <strong>contradiction<\/strong><\/p>\n<hr \/>\n<h2>\ud83d\udd39 <strong>4. Contingency (Sometimes True, Sometimes False)<\/strong><\/h2>\n<blockquote><p>A <strong>contingency<\/strong> is a compound statement that is <strong>true for some cases<\/strong> and <strong>false for others<\/strong>.<\/p><\/blockquote>\n<h3>\ud83d\udd04 Example:<\/h3>\n<blockquote><p>(P \u2227 Q)<\/p><\/blockquote>\n<table>\n<thead>\n<tr>\n<th>P<\/th>\n<th>Q<\/th>\n<th>P \u2227 Q<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2714\ufe0f Sometimes True \u2192 It\u2019s a <strong>contingency<\/strong><\/p>\n<hr \/>\n<h2>\ud83d\udd39 <strong>5. Satisfiable Statement<\/strong><\/h2>\n<blockquote><p>A formula is <strong>satisfiable<\/strong> if <strong>at least one<\/strong> combination of truth values makes the formula <strong>true<\/strong>.<\/p><\/blockquote>\n<p>Example: (P \u2228 Q)<\/p>\n<ul>\n<li>This is <strong>satisfiable<\/strong>, because in cases like P=T, Q=F \u2192 Output is T<\/li>\n<\/ul>\n<hr \/>\n<h2>\ud83d\udd39 <strong>6. Unsatisfiable Statement<\/strong><\/h2>\n<blockquote><p>A formula is <strong>unsatisfiable<\/strong> if <strong>no combination<\/strong> of values makes it <strong>true<\/strong> \u2192 same as <strong>contradiction<\/strong><\/p><\/blockquote>\n<p>Example: (P \u2227 \u00acP)<\/p>\n<ul>\n<li>Always false \u2192 <strong>unsatisfiable<\/strong><\/li>\n<\/ul>\n<hr \/>\n<h2>\ud83c\udfaf Summary Table<\/h2>\n<table>\n<thead>\n<tr>\n<th>Concept<\/th>\n<th>Definition<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Tautology<\/strong><\/td>\n<td>Always true<\/td>\n<td>P \u2228 \u00acP<\/td>\n<\/tr>\n<tr>\n<td><strong>Contradiction<\/strong><\/td>\n<td>Always false<\/td>\n<td>P \u2227 \u00acP<\/td>\n<\/tr>\n<tr>\n<td><strong>Contingency<\/strong><\/td>\n<td>Sometimes true, sometimes false<\/td>\n<td>P \u2192 Q<\/td>\n<\/tr>\n<tr>\n<td><strong>Satisfiable<\/strong><\/td>\n<td>At least one case is true<\/td>\n<td>P \u2228 Q<\/td>\n<\/tr>\n<tr>\n<td><strong>Unsatisfiable<\/strong><\/td>\n<td>No case is true<\/td>\n<td>P \u2227 \u00acP<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h2>\ud83d\udcdd Want Exercises or Truth Table Practice?<\/h2>\n<p>I can provide:<\/p>\n<ul>\n<li>\ud83d\udcc4 PDF summary notes<\/li>\n<li>\ud83e\udde0 Practice Questions with Truth Table<\/li>\n<li>\ud83c\udfa5 Hindi or English Video Explanation<\/li>\n<li>\ud83e\uddea Quiz (MCQs) on this topic<\/li>\n<\/ul>\n<p>Tell me what you&#8217;d like!<\/p>\n<h3><a href=\"https:\/\/faculty.ksu.edu.sa\/sites\/default\/files\/rosen_discrete_mathematics_and_its_applications_7th_edition.pdf\" target=\"_blank\" rel=\"noopener\">Day 04Part12-Concept of tautology contradiction and Contigency , Satisfiable and Unsatisfiable case.<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 04Part12-Concept of tautology contradiction and Contigency, Satisfiable and Unsatisfiable case. [fvplayer id=&#8221;201&#8243;] Day 04 &#8211; Part 12: Concept of Tautology, Contradiction, and Contingency In Discrete Mathematics and Logic, statements can be categorized based on their truth values in all possible scenarios. The key concepts are Tautology, Contradiction, and Contingency, along with Satisfiable and Unsatisfiable [&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-2983","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2983","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=2983"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2983\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2983"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2983"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}