Part 02- Properties of Proposition law of excluded middle and law of contradiction.<\/p>\n
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Here is Part 02<\/strong> of Discrete Mathematics \u2013 Properties of Proposition<\/em>, focusing on two fundamental logical laws: the Law of Excluded Middle<\/strong> and the Law of Contradiction<\/strong>. These are essential for understanding propositional logic<\/strong>, used in mathematics, computer science, and GATE CSE<\/strong>.<\/p>\n A proposition<\/strong> is a declarative statement<\/strong> that is either true (T)<\/strong> or false (F)<\/strong>, but not both.<\/p>\n Example:<\/p>\n \u2705 “2 + 2 = 4” (True)<\/p>\n<\/li>\n \u274c “It will rain today” (Not a proposition unless it has a definite truth value)<\/p>\n<\/li>\n<\/ul>\n For any proposition P<\/strong>:<\/p>\n P\u2228\u00acP\u00a0is\u00a0always\u00a0TRUEP \\lor \\neg P \\text{ is always TRUE}<\/span>P<\/span>\u2228<\/span><\/span>\u00ac<\/span>P<\/span>\u00a0is\u00a0always\u00a0TRUE<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n This means that either the proposition is true, or its negation is true<\/strong>. There’s no third option.<\/p>\n Let P = “5 is an odd number”<\/strong><\/p>\n P=TrueP = \\text{True}<\/span>P<\/span>=<\/span><\/span>True<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n \u00acP=False\\neg P = \\text{False}<\/span>\u00ac<\/span>P<\/span>=<\/span><\/span>False<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n P\u2228\u00acP=True\u2228False=TrueP \\lor \\neg P = \\text{True} \\lor \\text{False} = \\text{True}<\/span>P<\/span>\u2228<\/span><\/span>\u00ac<\/span>P<\/span>=<\/span><\/span>True<\/span><\/span>\u2228<\/span><\/span>False<\/span><\/span>=<\/span><\/span>True<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n \u2705 So the law holds.<\/p>\n Forms the basis of classical logic<\/strong>.<\/p>\n<\/li>\n Used in proof by contradiction<\/strong>.<\/p>\n<\/li>\n Helps in simplifying logical expressions in computer science.<\/p>\n<\/li>\n<\/ul>\n For any proposition P<\/strong>:<\/p>\n P\u2227\u00acP\u00a0is\u00a0always\u00a0FALSEP \\land \\neg P \\text{ is always FALSE}<\/span>P<\/span>\u2227<\/span><\/span>\u00ac<\/span>P<\/span>\u00a0is\u00a0always\u00a0FALSE<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n This means a proposition cannot be both true and false at the same time<\/strong>.<\/p>\n Let P = “The Earth is flat”<\/strong><\/p>\n P=FalseP = \\text{False}<\/span>P<\/span>=<\/span><\/span>False<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n \u00acP=True\\neg P = \\text{True}<\/span>\u00ac<\/span>P<\/span>=<\/span><\/span>True<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n P\u2227\u00acP=False\u2227True=FalseP \\land \\neg P = \\text{False} \\land \\text{True} = \\text{False}<\/span>P<\/span>\u2227<\/span><\/span>\u00ac<\/span>P<\/span>=<\/span><\/span>False<\/span><\/span>\u2227<\/span><\/span>True<\/span><\/span>=<\/span><\/span>False<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n \u2705 So the law holds.<\/p>\n Ensures logical consistency<\/strong><\/p>\n<\/li>\n Used to reject contradictory systems<\/strong><\/p>\n<\/li>\n Essential in automated theorem proving<\/strong> and program correctness<\/strong><\/p>\n<\/li>\n<\/ul>\n
\n\ud83e\udde0 What is a Proposition?<\/h2>\n
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\n\ud83d\udd39 Law of Excluded Middle (LEM)<\/h2>\n
\ud83d\udcd8 Definition:<\/strong><\/h3>\n
\u2705 Example:<\/strong><\/h3>\n
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\n\ud83e\udde0 Why It Matters:<\/strong><\/h3>\n
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\n\ud83d\udd38 Law of Contradiction<\/h2>\n
\ud83d\udcd8 Definition:<\/strong><\/h3>\n
\u2705 Example:<\/strong><\/h3>\n
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\n\ud83e\udde0 Why It Matters:<\/strong><\/h3>\n
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\n\ud83d\udcca Truth Table Comparison<\/h2>\n