{"id":2948,"date":"2025-06-06T07:13:13","date_gmt":"2025-06-06T07:13:13","guid":{"rendered":"https:\/\/diznr.com\/?p=2948"},"modified":"2025-06-06T07:13:13","modified_gmt":"2025-06-06T07:13:13","slug":"tautology-concept-of-one-gate-2020-previous-year-gate-question-paper-let-pqr-be-atomic-three","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/tautology-concept-of-one-gate-2020-previous-year-gate-question-paper-let-pqr-be-atomic-three\/","title":{"rendered":"Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic."},"content":{"rendered":"<p>Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic.<\/p>\n<p>[fvplayer id=&#8221;184&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"65\"><strong data-start=\"4\" data-end=\"63\">\u00a0Tautology Concept &#8211; GATE 2025 (Discrete Mathematics)<\/strong><\/h3>\n<h4 data-start=\"67\" data-end=\"101\"><strong data-start=\"72\" data-end=\"99\">\u00a0What is a Tautology?<\/strong><\/h4>\n<p data-start=\"102\" data-end=\"232\">A <strong data-start=\"104\" data-end=\"117\">tautology<\/strong> is a <strong data-start=\"123\" data-end=\"156\">propositional logic statement<\/strong> that is <strong data-start=\"165\" data-end=\"180\">always true<\/strong> regardless of the truth values of its components.<\/p>\n<p data-start=\"234\" data-end=\"424\"><strong data-start=\"236\" data-end=\"268\">Mathematical Representation:<\/strong><br data-start=\"268\" data-end=\"271\" \/>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 a logical proposition, then <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 a <strong data-start=\"326\" data-end=\"339\">tautology<\/strong> if its <strong data-start=\"347\" data-end=\"423\">truth table always evaluates to &#8220;True&#8221; (T) for all possible input values<\/strong>.<\/p>\n<p data-start=\"234\" data-end=\"424\"><strong data-start=\"435\" data-end=\"464\">\u00a0Example of a Tautology<\/strong><\/p>\n<ol data-start=\"467\" data-end=\"1028\">\n<li data-start=\"467\" data-end=\"842\">\n<p data-start=\"470\" data-end=\"509\"><strong data-start=\"470\" data-end=\"507\">Logical OR with a Truth Statement<\/strong><\/p>\n<p><span class=\"katex-display\"><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><\/span><\/p>\n<ul data-start=\"542\" data-end=\"842\">\n<li data-start=\"542\" data-end=\"573\">This means <strong data-start=\"555\" data-end=\"571\">&#8220;P OR NOT P&#8221;<\/strong><\/li>\n<li data-start=\"577\" data-end=\"657\">This is always true 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.<\/li>\n<li data-start=\"661\" data-end=\"805\">Example:\n<ul data-start=\"677\" data-end=\"805\">\n<li data-start=\"677\" data-end=\"737\">If <span class=\"katex\"><span class=\"katex-mathml\">P=TP = T<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">\u00acP=F\\neg P = F<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">F<\/span><\/span><\/span><\/span> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">T\u2228F=TT \\lor F = T<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">F<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"743\" data-end=\"805\">If <span class=\"katex\"><span class=\"katex-mathml\">P=FP = F<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">F<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">\u00acP=T\\neg P = T<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><\/span><\/span><\/span> \u2192 <span class=\"katex\"><span class=\"katex-mathml\">F\u2228T=TF \\lor T = T<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">F<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"809\" data-end=\"842\"><strong data-start=\"813\" data-end=\"840\">Always True \u2192 Tautology<\/strong><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"844\" data-end=\"1028\">\n<p data-start=\"847\" data-end=\"884\"><strong data-start=\"847\" data-end=\"882\">Implication Rule as a Tautology<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2228(\u00acP)(P \\rightarrow Q) \\lor (\\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=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<ul data-start=\"935\" data-end=\"1028\">\n<li data-start=\"935\" data-end=\"980\">This means <strong data-start=\"948\" data-end=\"978\">&#8220;If P implies Q, OR not P&#8221;<\/strong><\/li>\n<li data-start=\"984\" data-end=\"1028\">Always evaluates to True in a truth table.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 data-start=\"1035\" data-end=\"1093\"><strong data-start=\"1039\" data-end=\"1091\">\u00a0GATE 2025 Previous Year Question on Tautology<\/strong><\/h3>\n<p data-start=\"1094\" data-end=\"1161\">Let <span class=\"katex\"><span class=\"katex-mathml\">P,Q,RP, Q, R<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> be three atomic propositions. The statement is:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2228Q)\u2192(P\u2228Q\u2228R)(P \\lor Q) \\rightarrow (P \\lor Q \\lor R)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><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 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=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<h4 data-start=\"1212\" data-end=\"1247\"><strong data-start=\"1217\" data-end=\"1247\">Solution Using Truth Table<\/strong><\/h4>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"1248\" data-end=\"1603\">\n<thead data-start=\"1248\" data-end=\"1369\">\n<tr data-start=\"1248\" data-end=\"1369\">\n<th data-start=\"1248\" data-end=\"1258\"><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=\"1258\" data-end=\"1268\"><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=\"1268\" data-end=\"1278\"><span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><\/th>\n<th data-start=\"1278\" data-end=\"1295\"><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<th data-start=\"1295\" data-end=\"1319\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2228Q\u2228RP \\lor Q \\lor R<\/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 class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><\/th>\n<th data-start=\"1319\" data-end=\"1369\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2228Q)\u2192(P\u2228Q\u2228R)(P \\lor Q) \\rightarrow (P \\lor Q \\lor R)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><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 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=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1396\" data-end=\"1603\">\n<tr data-start=\"1396\" data-end=\"1421\">\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1422\" data-end=\"1447\">\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1448\" data-end=\"1473\">\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1474\" data-end=\"1499\">\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1500\" data-end=\"1525\">\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1526\" data-end=\"1551\">\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1552\" data-end=\"1577\">\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"1578\" data-end=\"1603\">\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p data-start=\"1605\" data-end=\"1682\">\u00a0The last column is <strong data-start=\"1626\" data-end=\"1641\">always TRUE<\/strong>, so this statement is a <strong data-start=\"1666\" data-end=\"1679\">tautology<\/strong>.<\/p>\n<h3 data-start=\"1689\" data-end=\"1729\"><strong data-start=\"1693\" data-end=\"1727\">\u00a0Key Takeaways for GATE 2025<\/strong><\/h3>\n<ol data-start=\"1730\" data-end=\"2078\">\n<li data-start=\"1730\" data-end=\"1816\"><strong data-start=\"1733\" data-end=\"1746\">Tautology<\/strong>: A statement that is <strong data-start=\"1768\" data-end=\"1783\">always true<\/strong> for all values of P, Q, and R.<\/li>\n<li data-start=\"1817\" data-end=\"1973\"><strong data-start=\"1820\" data-end=\"1846\">Common Tautology Forms<\/strong>:\n<ul data-start=\"1853\" data-end=\"1973\">\n<li data-start=\"1853\" data-end=\"1880\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2228\u00acP=TP \\lor \\neg P = T<\/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 class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">T<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1884\" data-end=\"1923\"><span class=\"katex\"><span class=\"katex-mathml\">(P\u2192Q)\u2228\u00acP(P \\rightarrow Q) \\lor \\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=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1927\" data-end=\"1973\"><span class=\"katex\"><span class=\"katex-mathml\">P\u2228Q\u2192P\u2228Q\u2228RP \\lor Q \\rightarrow P \\lor Q \\lor R<\/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 class=\"mrel\">\u2192<\/span><\/span><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 class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1974\" data-end=\"2078\"><strong data-start=\"1977\" data-end=\"1995\">GATE Questions<\/strong> often test tautology using <strong data-start=\"2023\" data-end=\"2075\">truth tables, logical identities, and set theory<\/strong>.<\/li>\n<\/ol>\n<p data-start=\"2085\" data-end=\"2163\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"2088\" data-end=\"2163\" data-is-last-node=\"\">Would you like more solved GATE questions on Tautology for practice?<\/strong><\/p>\n<h3 data-start=\"2085\" data-end=\"2163\"><a href=\"https:\/\/mrce.in\/ebooks\/Maths-Discrete%20Mathematics%20&amp;%20its%20Applications%208th%20Ed.pdf\" target=\"_blank\" rel=\"noopener\">Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic.<\/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\">Certainly! Let&#8217;s delve into the concept of <strong data-start=\"43\" data-end=\"56\">tautology<\/strong> in propositional logic, particularly in the context of GATE (Graduate Aptitude Test in Engineering) examinations.<\/span><\/p>\n<hr data-start=\"76\" data-end=\"79\" \/>\n<h3 data-start=\"81\" data-end=\"108\">\ud83d\udd39 What is a Tautology?<\/h3>\n<p data-start=\"110\" data-end=\"184\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">In propositional logic, a <strong data-start=\"26\" data-end=\"39\">tautology<\/strong> is a compound proposition that is always true, regardless of the truth values of its constituent variables. This means that no matter how you assign truth values to the individual propositions, the overall expression evaluates to true.<\/span><\/p>\n<p data-start=\"186\" data-end=\"198\"><strong data-start=\"186\" data-end=\"198\">Example:<\/strong><\/p>\n<p data-start=\"200\" data-end=\"318\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Consider the expression:<\/span><br data-start=\"237\" data-end=\"240\" \/><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(p\u2227q)\u2192(p\u2228q)(p \\land q) \\rightarrow (p \\lor 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><\/span><\/span><\/p>\n<p data-start=\"320\" data-end=\"398\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This is a tautology because:<\/span><\/p>\n<ul data-start=\"400\" data-end=\"522\">\n<li data-start=\"400\" data-end=\"441\">\n<p data-start=\"402\" data-end=\"441\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">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 <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> is also true, making the implication true.<\/span><\/p>\n<\/li>\n<li data-start=\"442\" data-end=\"522\">\n<p data-start=\"444\" data-end=\"522\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">If 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> or <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> is false, then <span class=\"katex\"><span class=\"katex-mathml\">p\u2227qp \\land q<\/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 mathnormal\">q<\/span><\/span><\/span><\/span> is false, and a false antecedent in an implication makes the entire implication true.<\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"524\" data-end=\"602\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Therefore, in all possible truth assignments, the expression evaluates to true.<\/span><\/p>\n<hr data-start=\"604\" data-end=\"607\" \/>\n<h3 data-start=\"609\" data-end=\"651\">\ud83d\udd39 GATE Previous Year Question Example<\/h3>\n<p data-start=\"653\" data-end=\"731\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Let&#8217;s analyze a question from <strong data-start=\"30\" data-end=\"52\">GATE CS 2017 Set 1<\/strong>:<\/span><\/p>\n<p data-start=\"733\" data-end=\"746\"><strong data-start=\"733\" data-end=\"746\">Question:<\/strong><\/p>\n<p data-start=\"748\" data-end=\"868\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Let <span class=\"katex\"><span class=\"katex-mathml\">p,q,p, q,<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">q<\/span><span class=\"mpunct\">,<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">rr<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><\/span><\/span><\/span> be propositions, and consider the expression:<\/span><br data-start=\"787\" data-end=\"790\" \/><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(p\u2192q)\u2192r(p \\rightarrow q) \\rightarrow r<\/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=\"mord mathnormal\">r<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"870\" data-end=\"990\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Suppose this expression is a <strong data-start=\"29\" data-end=\"46\">contradiction<\/strong>, meaning it is always false. Then, evaluate the nature of the expression:<\/span><br data-start=\"909\" data-end=\"912\" \/><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(r\u2192p)\u2192q(r \\rightarrow p) \\rightarrow q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">q<\/span><\/span><\/span><\/span><\/span><\/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<p data-start=\"992\" data-end=\"1005\"><strong data-start=\"992\" data-end=\"1005\">Solution:<\/strong><\/p>\n<ol data-start=\"1007\" data-end=\"2408\">\n<li data-start=\"1007\" data-end=\"1458\">\n<p data-start=\"1010\" data-end=\"1046\"><strong data-start=\"1010\" data-end=\"1046\">Understanding the Contradiction:<\/strong><\/p>\n<p data-start=\"1051\" data-end=\"1328\">The expression <span class=\"katex\"><span class=\"katex-mathml\">(p\u2192q)\u2192r(p \\rightarrow q) \\rightarrow r<\/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=\"mord mathnormal\">r<\/span><\/span><\/span><\/span> being a contradiction implies that it is false for all truth assignments. An implication <span class=\"katex\"><span class=\"katex-mathml\">A\u2192BA \\rightarrow B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> is false only when <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> is true and <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> is false. Therefore, for the entire expression to be always false:<\/p>\n<ul data-start=\"1333\" data-end=\"1458\">\n<li data-start=\"1333\" data-end=\"1374\">\n<p data-start=\"1335\" data-end=\"1374\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2192qp \\rightarrow q<\/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=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> must be true.<\/span><\/p>\n<\/li>\n<li data-start=\"1378\" data-end=\"1458\">\n<p data-start=\"1380\" data-end=\"1458\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">rr<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><\/span><\/span><\/span> must be false.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1460\" data-end=\"2408\">\n<p data-start=\"1463\" data-end=\"1515\"><strong data-start=\"1463\" data-end=\"1515\">Analyzing <span class=\"katex\"><span class=\"katex-mathml\">(r\u2192p)\u2192q(r \\rightarrow p) \\rightarrow q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">q<\/span><\/span><\/span><\/span>:<\/strong><\/p>\n<p data-start=\"1520\" data-end=\"1590\">Let&#8217;s examine the truth value of this expression under the conditions:<\/p>\n<ul data-start=\"1595\" data-end=\"1720\">\n<li data-start=\"1595\" data-end=\"1636\">\n<p data-start=\"1597\" data-end=\"1636\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">rr<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><\/span><\/span><\/span> is false.<\/span><\/p>\n<\/li>\n<li data-start=\"1640\" data-end=\"1720\">\n<p data-start=\"1642\" data-end=\"1720\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">p\u2192qp \\rightarrow q<\/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=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> is true.<\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1725\" data-end=\"1876\">Since <span class=\"katex\"><span class=\"katex-mathml\">rr<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><\/span><\/span><\/span> is false, <span class=\"katex\"><span class=\"katex-mathml\">r\u2192pr \\rightarrow p<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> is true (an implication with a false antecedent is always true). Therefore, the expression simplifies to:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">true\u2192q\\text{true} \\rightarrow q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">true<\/span><\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">q<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1923\" data-end=\"1978\">The truth value of this implication depends on <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>:<\/p>\n<ul data-start=\"1983\" data-end=\"2108\">\n<li data-start=\"1983\" data-end=\"2024\">\n<p data-start=\"1985\" data-end=\"2024\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">If <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> is true, then the implication is true.<\/span><\/p>\n<\/li>\n<li data-start=\"2028\" data-end=\"2108\">\n<p data-start=\"2030\" data-end=\"2108\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">If <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> is false, then the implication is false.<\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2113\" data-end=\"2323\">However, since <span class=\"katex\"><span class=\"katex-mathml\">p\u2192qp \\rightarrow q<\/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=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> is true and <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> could be true or false, the only way <span class=\"katex\"><span class=\"katex-mathml\">p\u2192qp \\rightarrow q<\/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=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> is always true is if <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> is true. Therefore, under these conditions, <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> must be true.<\/p>\n<p data-start=\"2328\" data-end=\"2408\">Thus, <span class=\"katex\"><span class=\"katex-mathml\">(r\u2192p)\u2192q(r \\rightarrow p) \\rightarrow q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> is always true when <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> is true.<\/p>\n<\/li>\n<\/ol>\n<p data-start=\"2410\" data-end=\"2500\"><strong data-start=\"2410\" data-end=\"2421\">Answer:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">The expression <span class=\"katex\"><span class=\"katex-mathml\">(r\u2192p)\u2192q(r \\rightarrow p) \\rightarrow q<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">r<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">q<\/span><\/span><\/span><\/span> is <strong data-start=\"56\" data-end=\"92\">always true when <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> is true<\/strong>.<\/span><\/p>\n<hr data-start=\"2502\" data-end=\"2505\" \/>\n<h3 data-start=\"2507\" data-end=\"2534\">\ud83d\udd39 Additional Resources<\/h3>\n<p data-start=\"2536\" data-end=\"2635\">For a more in-depth understanding and further practice, consider exploring the following resources:<\/p>\n<ul data-start=\"2637\" data-end=\"3392\">\n<li data-start=\"2637\" data-end=\"2907\">\n<p data-start=\"2639\" data-end=\"2907\"><strong data-start=\"2639\" data-end=\"2688\">GATE Overflow \u2013 Mathematical Logic Questions:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A comprehensive collection of previous GATE questions on mathematical logic.<\/span> (gateoverflow.in)<\/p>\n<\/li>\n<li data-start=\"2909\" data-end=\"3181\">\n<p data-start=\"2911\" data-end=\"3181\"><strong data-start=\"2911\" data-end=\"2957\">GeeksforGeeks \u2013 Propositional Logic Notes:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Detailed notes on propositional and first-order logic, including tautologies and other logical connectives.<\/span> (geeksforgeeks.org)<\/p>\n<\/li>\n<li data-start=\"3183\" data-end=\"3392\">\n<p data-start=\"3185\" data-end=\"3392\"><strong data-start=\"3185\" data-end=\"3229\">YouTube \u2013 Propositional Logic GATE PYQs:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A video explanation of previous year GATE questions on propositional logic.<\/span> (youtube.com)<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"3394\" data-end=\"3397\" \/>\n<p data-start=\"3399\" data-end=\"3495\">If you have any specific questions or need further clarification on any topic, feel free to ask!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic. [fvplayer id=&#8221;184&#8243;] \u00a0Tautology Concept &#8211; GATE 2025 (Discrete Mathematics) \u00a0What is a Tautology? A tautology is a propositional logic statement that is always true regardless of the truth values of its components. Mathematical Representation:If PPP is a logical proposition, [&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-2948","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2948","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=2948"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2948\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2948"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2948"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}