Day 04Part 06- Discrete mathematics for computer science – Dis-junction Operator of proposition.<\/p>\n
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In propositional logic<\/strong>, the disjunction operator (\u2228)<\/strong> represents the logical “OR”<\/strong> operation. It is used to form compound statements that are true if at least one of the given propositions is true<\/strong>.<\/p>\n The disjunction (P \u2228 Q)<\/strong> of two propositions P<\/strong> and Q<\/strong> is true if either P is true, Q is true, or both are true<\/strong>.<\/p>\n<\/li>\n The only case when P \u2228 Q is false<\/strong> is when both P and Q are false<\/strong>.<\/p>\n<\/li>\n<\/ul>\n Truth Table for Disjunction (P \u2228 Q):<\/strong><\/p>\n P \u2228 Q<\/strong> reads as “P OR Q<\/strong>“<\/p>\n<\/li>\n Example:<\/p>\n P<\/strong> = “It is raining.”<\/p>\n<\/li>\n Q<\/strong> = “It is cloudy.”<\/p>\n<\/li>\n P \u2228 Q<\/strong> = “It is raining OR it is cloudy.”<\/p>\n<\/li>\n The statement is true<\/strong> if at least one of the conditions is true.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n P\u2228Q=Q\u2228PP \u2228 Q = Q \u2228 P<\/span>P<\/span>\u2228<\/span><\/span>Q<\/span>=<\/span><\/span>Q<\/span>\u2228<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span><\/p>\n \u00a0Example: “It is raining OR it is cloudy” is the same as “It is cloudy OR it is raining.”<\/p>\n (P\u2228Q)\u2228R=P\u2228(Q\u2228R)(P \u2228 Q) \u2228 R = P \u2228 (Q \u2228 R)<\/span>(<\/span>P<\/span>\u2228<\/span><\/span>Q<\/span>)<\/span>\u2228<\/span><\/span>R<\/span>=<\/span><\/span>P<\/span>\u2228<\/span><\/span>(<\/span>Q<\/span>\u2228<\/span><\/span>R<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n \u00a0Example: If P = “Hot”<\/strong>, Q = “Cold”<\/strong>, and R = “Windy”<\/strong>, the order in which we group them doesn\u2019t change the truth.<\/p>\n P\u2228False=PP \u2228 \\text{False} = P<\/span>P<\/span>\u2228<\/span><\/span>False<\/span><\/span>=<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span><\/p>\n \u00a0Example: “It is sunny OR False” \u2192 The truth value remains that of “It is sunny.”<\/p>\n P\u2228True=TrueP \u2228 \\text{True} = \\text{True}<\/span>P<\/span>\u2228<\/span><\/span>True<\/span><\/span>=<\/span><\/span>True<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n \u00a0Example: “It is sunny OR True” \u2192 Always True.<\/p>\n P\u2228P=PP \u2228 P = P<\/span>P<\/span>\u2228<\/span><\/span>P<\/span>=<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span><\/p>\n \u00a0Example: “It is raining OR it is raining” \u2192 No change in truth value.<\/p>\n P\u2228(Q\u2227R)=(P\u2228Q)\u2227(P\u2228R)P \u2228 (Q \u2227 R) = (P \u2228 Q) \u2227 (P \u2228 R)<\/span>P<\/span>\u2228<\/span><\/span>(<\/span>Q<\/span>\u2227<\/span><\/span>R<\/span>)<\/span>=<\/span><\/span>(<\/span>P<\/span>\u2228<\/span><\/span>Q<\/span>)<\/span>\u2227<\/span><\/span>(<\/span>P<\/span>\u2228<\/span><\/span>R<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n Boolean Algebra<\/strong>: Used in circuit design and logic gates.<\/p>\n<\/li>\n Programming<\/strong>: Conditional statements (e.g., Database Queries<\/strong>: SQL uses Artificial Intelligence & Machine Learning<\/strong>: Logical operations in rule-based systems.<\/p>\n<\/li>\n<\/ul>\n Implemented as an “OR” gate<\/strong> in digital circuits.<\/p>\n<\/li>\n Output is 1 (True)<\/strong> if at least one input is 1<\/strong>.<\/p>\n<\/li>\n<\/ul>\n1. Definition of Disjunction (\u2228)<\/strong><\/h3>\n
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\n \nP<\/th>\n Q<\/th>\n P \u2228 Q<\/th>\n<\/tr>\n<\/thead>\n \n T<\/td>\n T<\/td>\n T<\/td>\n<\/tr>\n \n T<\/td>\n F<\/td>\n T<\/td>\n<\/tr>\n \n F<\/td>\n T<\/td>\n T<\/td>\n<\/tr>\n \n F<\/td>\n F<\/td>\n F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n 2. Symbolic Representation<\/strong><\/h3>\n
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3. Key Properties of Disjunction (\u2228)<\/strong><\/h3>\n
(A) Commutative Property:<\/strong><\/h3>\n
(B) Associative Property:<\/strong><\/h3>\n
(C) Identity Law:<\/strong><\/h3>\n
(D) Domination Law:<\/strong><\/h3>\n
(E) Idempotent Law:<\/strong><\/h3>\n
(F) Distributive Law:<\/strong><\/h3>\n
4. Application in Computer Science<\/strong><\/h3>\n
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if (A || B) {}<\/code> in C, Java, Python).<\/p>\n<\/li>\nOR<\/code> to filter data (SELECT * FROM students WHERE grade = 'A' OR attendance > 80;<\/code>).<\/p>\n<\/li>\n5. Disjunction in Logic Gates<\/strong><\/h3>\n
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