{"id":2989,"date":"2025-06-09T14:23:21","date_gmt":"2025-06-09T14:23:21","guid":{"rendered":"https:\/\/diznr.com\/?p=2989"},"modified":"2025-06-09T14:23:21","modified_gmt":"2025-06-09T14:23:21","slug":"day-04part-09-discrete-mathematics-for-computer-science-example-implication-statement-conditional","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-04part-09-discrete-mathematics-for-computer-science-example-implication-statement-conditional\/","title":{"rendered":"Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement."},"content":{"rendered":"<p>Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.<\/p>\n<p>[fvplayer id=&#8221;204&#8243;]<\/p>\n<h3 class=\"\" data-start=\"0\" data-end=\"69\"><strong data-start=\"4\" data-end=\"67\">Implication (Conditional Statement) in Discrete Mathematics<\/strong><\/h3>\n<h4 class=\"\" data-start=\"71\" data-end=\"93\"><strong data-start=\"76\" data-end=\"91\">Definition:<\/strong><\/h4>\n<p class=\"\" data-start=\"94\" data-end=\"192\">An <strong data-start=\"97\" data-end=\"112\">implication<\/strong> (also called a <strong data-start=\"128\" data-end=\"153\">conditional statement<\/strong>) is a logical statement of the form:<\/p>\n<p><span class=\"katex-display\"><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><\/span><\/p>\n<p class=\"\" data-start=\"217\" data-end=\"276\">which is read as <strong data-start=\"234\" data-end=\"252\">&#8220;If P, then Q&#8221;<\/strong> or <strong data-start=\"256\" data-end=\"273\">&#8220;P implies Q&#8221;<\/strong>.<\/p>\n<ul data-start=\"278\" data-end=\"398\">\n<li class=\"\" data-start=\"278\" data-end=\"343\">\n<p class=\"\" data-start=\"280\" data-end=\"343\"><strong data-start=\"280\" data-end=\"341\"><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 called the antecedent (hypothesis or premise).<\/strong><\/p>\n<\/li>\n<li class=\"\" data-start=\"344\" data-end=\"398\">\n<p class=\"\" data-start=\"346\" data-end=\"398\"><strong data-start=\"346\" data-end=\"396\"><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 called the consequent (conclusion).<\/strong><\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"405\" data-end=\"442\"><strong data-start=\"409\" data-end=\"440\">Truth Table for Implication<\/strong><\/h3>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"444\" data-end=\"679\">\n<thead data-start=\"444\" data-end=\"489\">\n<tr data-start=\"444\" data-end=\"489\">\n<th data-start=\"444\" data-end=\"454\"><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=\"454\" data-end=\"464\"><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=\"464\" data-end=\"489\"><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><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"528\" data-end=\"679\">\n<tr data-start=\"528\" data-end=\"565\">\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"566\" data-end=\"603\">\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr data-start=\"604\" data-end=\"641\">\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr data-start=\"642\" data-end=\"679\">\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p class=\"\" data-start=\"681\" data-end=\"707\"><strong data-start=\"684\" data-end=\"705\">Key Observations:<\/strong><\/p>\n<ul data-start=\"708\" data-end=\"865\">\n<li class=\"\" data-start=\"708\" data-end=\"779\">\n<p class=\"\" data-start=\"710\" data-end=\"779\">An implication is <strong data-start=\"728\" data-end=\"737\">false<\/strong> only when <strong data-start=\"748\" data-end=\"776\">P is true and Q is false<\/strong>.<\/p>\n<\/li>\n<li class=\"\" data-start=\"780\" data-end=\"865\">\n<p class=\"\" data-start=\"782\" data-end=\"865\">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 <strong data-start=\"796\" data-end=\"805\">false<\/strong>, the statement is <strong data-start=\"824\" data-end=\"839\">always true<\/strong>, 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>.<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"872\" data-end=\"916\"><strong data-start=\"876\" data-end=\"914\">Examples of Implication Statements<\/strong><\/h3>\n<h4 class=\"\" data-start=\"918\" data-end=\"967\"><strong data-start=\"923\" data-end=\"965\">Example 1: Basic Conditional Statement<\/strong><\/h4>\n<p class=\"\" data-start=\"968\" data-end=\"1031\"><strong data-start=\"971\" data-end=\"985\">Statement:<\/strong> <em data-start=\"986\" data-end=\"1029\">If it rains, then the ground will be wet.<\/em><\/p>\n<ul data-start=\"1032\" data-end=\"1126\">\n<li class=\"\" data-start=\"1032\" data-end=\"1054\">\n<p class=\"\" data-start=\"1034\" data-end=\"1054\"><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>: It rains.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1055\" data-end=\"1086\">\n<p class=\"\" data-start=\"1057\" data-end=\"1086\"><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>: The ground is wet.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1087\" data-end=\"1126\">\n<p class=\"\" data-start=\"1089\" data-end=\"1126\">Logical form: <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><\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1128\" data-end=\"1174\"><strong data-start=\"1133\" data-end=\"1172\">Example 2: Mathematical Implication<\/strong><\/h4>\n<p class=\"\" data-start=\"1175\" data-end=\"1244\"><strong data-start=\"1178\" data-end=\"1192\">Statement:<\/strong> <em data-start=\"1193\" data-end=\"1242\">If a number is divisible by 4, then it is even.<\/em><\/p>\n<ul data-start=\"1245\" data-end=\"1424\">\n<li class=\"\" data-start=\"1245\" data-end=\"1285\">\n<p class=\"\" data-start=\"1247\" data-end=\"1285\"><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>: A number is divisible by 4.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1286\" data-end=\"1318\">\n<p class=\"\" data-start=\"1288\" data-end=\"1318\"><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>: The number is even.<\/p>\n<\/li>\n<li class=\"\" data-start=\"1319\" data-end=\"1358\">\n<p class=\"\" data-start=\"1321\" data-end=\"1358\">Logical form: <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><\/p>\n<\/li>\n<li class=\"\" data-start=\"1359\" data-end=\"1424\">\n<p class=\"\" data-start=\"1361\" data-end=\"1424\"><strong data-start=\"1363\" data-end=\"1371\">True<\/strong>, since every number divisible by 4 is always even.<\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1426\" data-end=\"1465\"><strong data-start=\"1431\" data-end=\"1463\">Example 3: False Implication<\/strong><\/h4>\n<p class=\"\" data-start=\"1466\" data-end=\"1527\"><strong data-start=\"1469\" data-end=\"1483\">Statement:<\/strong> <em data-start=\"1484\" data-end=\"1525\">If 2 is greater than 3, then 5 is even.<\/em><\/p>\n<ul data-start=\"1528\" data-end=\"1693\">\n<li class=\"\" data-start=\"1528\" data-end=\"1554\">\n<p class=\"\" data-start=\"1530\" data-end=\"1554\"><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>: 2 &gt; 3 (False)<\/p>\n<\/li>\n<li class=\"\" data-start=\"1555\" data-end=\"1585\">\n<p class=\"\" data-start=\"1557\" data-end=\"1585\"><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>: 5 is even (False)<\/p>\n<\/li>\n<li class=\"\" data-start=\"1586\" data-end=\"1693\">\n<p class=\"\" data-start=\"1588\" data-end=\"1693\">Since <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 <strong data-start=\"1605\" data-end=\"1614\">false<\/strong>, the statement <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 <strong data-start=\"1655\" data-end=\"1663\">true<\/strong> (based on the truth table).<\/p>\n<\/li>\n<\/ul>\n<h3 class=\"\" data-start=\"1700\" data-end=\"1747\"><strong data-start=\"1704\" data-end=\"1745\">Contrapositive, Converse, and Inverse<\/strong><\/h3>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"1749\" data-end=\"2059\">\n<thead data-start=\"1749\" data-end=\"1769\">\n<tr data-start=\"1749\" data-end=\"1769\">\n<th data-start=\"1749\" data-end=\"1756\">Form<\/th>\n<th data-start=\"1756\" data-end=\"1769\">Statement<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1792\" data-end=\"2059\">\n<tr data-start=\"1792\" data-end=\"1850\">\n<td><strong data-start=\"1794\" data-end=\"1809\">Implication<\/strong><\/td>\n<td><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> (If P, then Q)<\/td>\n<\/tr>\n<tr data-start=\"1851\" data-end=\"1906\">\n<td><strong data-start=\"1853\" data-end=\"1865\">Converse<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">Q\u2192PQ \\rightarrow P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><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><\/span><\/span> (If Q, then P)<\/td>\n<\/tr>\n<tr data-start=\"1907\" data-end=\"1979\">\n<td><strong data-start=\"1909\" data-end=\"1920\">Inverse<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">\u00acP\u2192\u00acQ\\neg P \\rightarrow \\neg Q<\/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\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord\">\u00ac<\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span> (If not P, then not Q)<\/td>\n<\/tr>\n<tr data-start=\"1980\" data-end=\"2059\">\n<td><strong data-start=\"1982\" data-end=\"2000\">Contrapositive<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">\u00acQ\u2192\u00acP\\neg Q \\rightarrow \\neg P<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><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><\/span><\/span> (If not Q, then not P)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p class=\"\" data-start=\"2061\" data-end=\"2148\"><strong data-start=\"2064\" data-end=\"2082\">Contrapositive<\/strong> is <strong data-start=\"2086\" data-end=\"2117\">always logically equivalent<\/strong> to the original implication.<\/p>\n<h3 class=\"\" data-start=\"2155\" data-end=\"2175\"><strong data-start=\"2159\" data-end=\"2173\">Conclusion<\/strong><\/h3>\n<p class=\"\" data-start=\"2176\" data-end=\"2370\">Implication (<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 one of the <strong data-start=\"2226\" data-end=\"2261\">fundamental logical connectives<\/strong> in discrete mathematics. It is widely used in <strong data-start=\"2308\" data-end=\"2370\" data-is-last-node=\"\">mathematical proofs, computer science, and logic circuits.<\/strong><\/p>\n<h3 data-start=\"2176\" data-end=\"2370\"><a href=\"https:\/\/www2.cs.uh.edu\/~arjun\/courses\/ds\/DiscMaths4CompSc.pdf\" target=\"_blank\" rel=\"noopener\">Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/courses.cs.duke.edu\/spring09\/cps102\/Lectures\/Book.pdf\" target=\"_blank\" rel=\"noopener\">DISCRETE MATHEMATICS FOR COMPUTER SCIENCE<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/unidel.edu.ng\/focelibrary\/books\/Discrete%20Mathematics%20for%20Computer%20Science%20(Pomde%20N.)%20(Z-Library).pdf\" target=\"_blank\" rel=\"noopener\">Discrete Mathematics for Computer Science<\/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 data-start=\"0\" data-end=\"168\">Here&#8217;s a clear and simple explanation of <strong data-start=\"41\" data-end=\"81\">Implication (Conditional Statements)<\/strong> with <strong data-start=\"87\" data-end=\"99\">examples<\/strong> from <strong data-start=\"105\" data-end=\"167\">Day 04 Part 09 &#8211; Discrete Mathematics for Computer Science<\/strong>.<\/p>\n<hr data-start=\"170\" data-end=\"173\" \/>\n<h2 data-start=\"175\" data-end=\"244\">\ud83d\udcd8 <strong data-start=\"181\" data-end=\"244\">Implication in Discrete Mathematics (Conditional Statement)<\/strong><\/h2>\n<h3 data-start=\"246\" data-end=\"275\">\u2705 What is an Implication?<\/h3>\n<p data-start=\"277\" data-end=\"341\">An <strong data-start=\"280\" data-end=\"295\">implication<\/strong> is a <strong data-start=\"301\" data-end=\"322\">logical statement<\/strong> that has the form:<\/p>\n<blockquote data-start=\"343\" data-end=\"391\">\n<p data-start=\"345\" data-end=\"391\"><strong data-start=\"345\" data-end=\"363\">&#8220;If p, then q&#8221;<\/strong><br data-start=\"363\" data-end=\"366\" \/>Symbolically:\u2003<strong data-start=\"382\" data-end=\"391\">p \u2192 q<\/strong><\/p>\n<\/blockquote>\n<ul data-start=\"393\" data-end=\"496\">\n<li data-start=\"393\" data-end=\"444\">\n<p data-start=\"395\" data-end=\"444\"><code data-start=\"395\" data-end=\"398\">p<\/code> is called the <strong data-start=\"413\" data-end=\"427\">antecedent<\/strong> (or hypothesis).<\/p>\n<\/li>\n<li data-start=\"445\" data-end=\"496\">\n<p data-start=\"447\" data-end=\"496\"><code data-start=\"447\" data-end=\"450\">q<\/code> is called the <strong data-start=\"465\" data-end=\"479\">consequent<\/strong> (or conclusion).<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"498\" data-end=\"501\" \/>\n<h3 data-start=\"503\" data-end=\"548\">\ud83d\udcd6 <strong data-start=\"510\" data-end=\"548\">Truth Table of Implication (p \u2192 q)<\/strong><\/h3>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"550\" data-end=\"897\">\n<thead data-start=\"550\" data-end=\"607\">\n<tr data-start=\"550\" data-end=\"607\">\n<th data-start=\"550\" data-end=\"567\" data-col-size=\"sm\">p (Hypothesis)<\/th>\n<th data-start=\"567\" data-end=\"584\" data-col-size=\"sm\">q (Conclusion)<\/th>\n<th data-start=\"584\" data-end=\"607\" data-col-size=\"sm\">p \u2192 q (Implication)<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"666\" data-end=\"897\">\n<tr data-start=\"666\" data-end=\"723\">\n<td data-start=\"666\" data-end=\"683\" data-col-size=\"sm\">T<\/td>\n<td data-col-size=\"sm\" data-start=\"683\" data-end=\"700\">T<\/td>\n<td data-col-size=\"sm\" data-start=\"700\" data-end=\"723\">T<\/td>\n<\/tr>\n<tr data-start=\"724\" data-end=\"781\">\n<td data-start=\"724\" data-end=\"741\" data-col-size=\"sm\">T<\/td>\n<td data-start=\"741\" data-end=\"758\" data-col-size=\"sm\">F<\/td>\n<td data-start=\"758\" data-end=\"781\" data-col-size=\"sm\"><strong data-start=\"760\" data-end=\"765\">F<\/strong> \u274c<\/td>\n<\/tr>\n<tr data-start=\"782\" data-end=\"839\">\n<td data-start=\"782\" data-end=\"799\" data-col-size=\"sm\">F<\/td>\n<td data-col-size=\"sm\" data-start=\"799\" data-end=\"816\">T<\/td>\n<td data-col-size=\"sm\" data-start=\"816\" data-end=\"839\">T<\/td>\n<\/tr>\n<tr data-start=\"840\" data-end=\"897\">\n<td data-start=\"840\" data-end=\"857\" data-col-size=\"sm\">F<\/td>\n<td data-col-size=\"sm\" data-start=\"857\" data-end=\"874\">F<\/td>\n<td data-col-size=\"sm\" data-start=\"874\" data-end=\"897\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<blockquote data-start=\"899\" data-end=\"980\">\n<p data-start=\"901\" data-end=\"980\">\ud83d\udd3a <strong data-start=\"904\" data-end=\"918\">Only false<\/strong> when the first part (p) is true and second part (q) is false.<\/p>\n<\/blockquote>\n<hr data-start=\"982\" data-end=\"985\" \/>\n<h3 data-start=\"987\" data-end=\"1024\">\ud83e\udde0 <strong data-start=\"994\" data-end=\"1024\">How to Read an Implication<\/strong><\/h3>\n<ul data-start=\"1026\" data-end=\"1172\">\n<li data-start=\"1026\" data-end=\"1083\">\n<p data-start=\"1028\" data-end=\"1083\"><strong data-start=\"1028\" data-end=\"1037\">p \u2192 q<\/strong> means <strong data-start=\"1044\" data-end=\"1083\">\u201cIf p happens, then q will happen.\u201d<\/strong><\/p>\n<\/li>\n<li data-start=\"1084\" data-end=\"1172\">\n<p data-start=\"1086\" data-end=\"1096\">Example:<\/p>\n<ul data-start=\"1099\" data-end=\"1172\">\n<li data-start=\"1099\" data-end=\"1172\">\n<p data-start=\"1101\" data-end=\"1172\">\u201cIf it rains, then the ground will be wet.\u201d<br data-start=\"1144\" data-end=\"1147\" \/>\u2192 <code data-start=\"1153\" data-end=\"1172\">Rain \u2192 Wet Ground<\/code><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr data-start=\"1174\" data-end=\"1177\" \/>\n<h2 data-start=\"1179\" data-end=\"1208\">\ud83e\uddea Examples of Implication<\/h2>\n<h3 data-start=\"1210\" data-end=\"1227\">\ud83d\udd39 Example 1:<\/h3>\n<blockquote data-start=\"1228\" data-end=\"1281\">\n<p data-start=\"1230\" data-end=\"1281\"><strong data-start=\"1230\" data-end=\"1281\">If a number is divisible by 4, then it is even.<\/strong><\/p>\n<\/blockquote>\n<ul data-start=\"1283\" data-end=\"1369\">\n<li data-start=\"1283\" data-end=\"1317\">\n<p data-start=\"1285\" data-end=\"1317\">p: A number is divisible by 4.<\/p>\n<\/li>\n<li data-start=\"1318\" data-end=\"1344\">\n<p data-start=\"1320\" data-end=\"1344\">q: The number is even.<\/p>\n<\/li>\n<li data-start=\"1345\" data-end=\"1369\">\n<p data-start=\"1347\" data-end=\"1369\">Implication: <strong data-start=\"1360\" data-end=\"1369\">p \u2192 q<\/strong><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1371\" data-end=\"1448\">\u2705 This is a <strong data-start=\"1383\" data-end=\"1391\">true<\/strong> implication because every number divisible by 4 is even.<\/p>\n<hr data-start=\"1450\" data-end=\"1453\" \/>\n<h3 data-start=\"1455\" data-end=\"1472\">\ud83d\udd39 Example 2:<\/h3>\n<blockquote data-start=\"1473\" data-end=\"1501\">\n<p data-start=\"1475\" data-end=\"1501\"><strong data-start=\"1475\" data-end=\"1501\">If x &gt; 5, then x\u00b2 &gt; 25<\/strong><\/p>\n<\/blockquote>\n<ul data-start=\"1503\" data-end=\"1626\">\n<li data-start=\"1503\" data-end=\"1626\">\n<p data-start=\"1505\" data-end=\"1516\">Let\u2019s test:<\/p>\n<ul data-start=\"1519\" data-end=\"1626\">\n<li data-start=\"1519\" data-end=\"1545\">\n<p data-start=\"1521\" data-end=\"1545\">x = 6 \u2192 x\u00b2 = 36 \u2192 \u2705 True<\/p>\n<\/li>\n<li data-start=\"1548\" data-end=\"1623\">\n<p data-start=\"1550\" data-end=\"1623\">x = 4 \u2192 x &gt; 5 is <strong data-start=\"1567\" data-end=\"1576\">false<\/strong> \u2192 So implication is <strong data-start=\"1597\" data-end=\"1605\">true<\/strong> (doesn\u2019t violate)<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p data-start=\"1627\" data-end=\"1672\">\u2705 Hence, the statement is <strong data-start=\"1653\" data-end=\"1671\">logically true<\/strong>.<\/p>\n<hr data-start=\"1674\" data-end=\"1677\" \/>\n<h3 data-start=\"1679\" data-end=\"1696\">\ud83d\udd39 Example 3:<\/h3>\n<blockquote data-start=\"1697\" data-end=\"1731\">\n<p data-start=\"1699\" data-end=\"1731\"><strong data-start=\"1699\" data-end=\"1731\">If 2 is odd, then 3 is even.<\/strong><\/p>\n<\/blockquote>\n<ul data-start=\"1733\" data-end=\"1830\">\n<li data-start=\"1733\" data-end=\"1758\">\n<p data-start=\"1735\" data-end=\"1758\">p: 2 is odd \u2192 \u274c False<\/p>\n<\/li>\n<li data-start=\"1759\" data-end=\"1785\">\n<p data-start=\"1761\" data-end=\"1785\">q: 3 is even \u2192 \u274c False<\/p>\n<\/li>\n<li data-start=\"1786\" data-end=\"1830\">\n<p data-start=\"1788\" data-end=\"1830\">So p \u2192 q = \u2705 <strong data-start=\"1801\" data-end=\"1809\">True<\/strong> (because p is false)<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1832\" data-end=\"1923\">\u26d4 Even though both are wrong, the implication is <strong data-start=\"1881\" data-end=\"1899\">logically true<\/strong> in propositional logic.<\/p>\n<hr data-start=\"1925\" data-end=\"1928\" \/>\n<h3 data-start=\"1930\" data-end=\"1980\">\ud83d\udd01 <strong data-start=\"1937\" data-end=\"1980\">Contrapositive and Converse (Advanced):<\/strong><\/h3>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1982\" data-end=\"2328\">\n<thead data-start=\"1982\" data-end=\"2039\">\n<tr data-start=\"1982\" data-end=\"2039\">\n<th data-start=\"1982\" data-end=\"2006\" data-col-size=\"sm\">Form<\/th>\n<th data-start=\"2006\" data-end=\"2039\" data-col-size=\"sm\">Statement<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2098\" data-end=\"2328\">\n<tr data-start=\"2098\" data-end=\"2155\">\n<td data-start=\"2098\" data-end=\"2122\" data-col-size=\"sm\">Implication (p \u2192 q)<\/td>\n<td data-col-size=\"sm\" data-start=\"2122\" data-end=\"2155\">If p, then q<\/td>\n<\/tr>\n<tr data-start=\"2156\" data-end=\"2213\">\n<td data-start=\"2156\" data-end=\"2180\" data-col-size=\"sm\">Converse (q \u2192 p)<\/td>\n<td data-col-size=\"sm\" data-start=\"2180\" data-end=\"2213\">If q, then p<\/td>\n<\/tr>\n<tr data-start=\"2214\" data-end=\"2271\">\n<td data-start=\"2214\" data-end=\"2238\" data-col-size=\"sm\">Inverse (\u00acp \u2192 \u00acq)<\/td>\n<td data-col-size=\"sm\" data-start=\"2238\" data-end=\"2271\">If not p, then not q<\/td>\n<\/tr>\n<tr data-start=\"2272\" data-end=\"2328\">\n<td data-start=\"2272\" data-end=\"2299\" data-col-size=\"sm\">Contrapositive (\u00acq \u2192 \u00acp)<\/td>\n<td data-col-size=\"sm\" data-start=\"2299\" data-end=\"2328\">If not q, then not p<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<blockquote data-start=\"2330\" data-end=\"2408\">\n<p data-start=\"2332\" data-end=\"2408\">Only <strong data-start=\"2337\" data-end=\"2355\">contrapositive<\/strong> is logically equivalent to the original implication.<\/p>\n<\/blockquote>\n<hr data-start=\"2410\" data-end=\"2413\" \/>\n<h2 data-start=\"2415\" data-end=\"2429\">\ud83d\udccc Summary:<\/h2>\n<ul data-start=\"2431\" data-end=\"2647\">\n<li data-start=\"2431\" data-end=\"2497\">\n<p data-start=\"2433\" data-end=\"2497\"><strong data-start=\"2433\" data-end=\"2448\">Implication<\/strong> expresses a condition: <strong data-start=\"2472\" data-end=\"2497\">\u201cIf this, then that.\u201d<\/strong><\/p>\n<\/li>\n<li data-start=\"2498\" data-end=\"2568\">\n<p data-start=\"2500\" data-end=\"2568\"><strong data-start=\"2500\" data-end=\"2519\">False only when<\/strong> the first part is true, and the second is false.<\/p>\n<\/li>\n<li data-start=\"2569\" data-end=\"2647\">\n<p data-start=\"2571\" data-end=\"2647\">Used widely in <strong data-start=\"2586\" data-end=\"2609\">mathematical proofs<\/strong>, <strong data-start=\"2611\" data-end=\"2625\">algorithms<\/strong>, and <strong data-start=\"2631\" data-end=\"2646\">logic gates<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2649\" data-end=\"2652\" \/>\n<p data-start=\"2654\" data-end=\"2768\" data-is-last-node=\"\" data-is-only-node=\"\">Let me know if you want <strong data-start=\"2678\" data-end=\"2699\">practice problems<\/strong>, <strong data-start=\"2701\" data-end=\"2726\">truth table exercises<\/strong>, or a <strong data-start=\"2733\" data-end=\"2752\">PDF notes sheet<\/strong> for this topic!<\/p>\n<h3 data-start=\"2654\" data-end=\"2768\"><a href=\"https:\/\/mrcet.com\/downloads\/digital_notes\/IT\/CSE%20_(R22)_2-2_DM%20DIGITAL%20NOTES.pdf\" target=\"_blank\" rel=\"noopener\">Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement. [fvplayer id=&#8221;204&#8243;] Implication (Conditional Statement) in Discrete Mathematics Definition: An implication (also called a conditional statement) is a logical statement of the form: P\u2192QP \\rightarrow QP\u2192Q which is read as &#8220;If P, then Q&#8221; or &#8220;P implies Q&#8221;. PPP is called the antecedent (hypothesis or [&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-2989","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2989","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=2989"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2989\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2989"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}