Mathematical Logic

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A statement is a declarative sentence which is either true or false, but not both simultaneously. Example: '2 is a prime number' (True), 'The Sun rises in the West' (False). Note: Questions, exclamati…

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A statement has a definite truth value (T or F), while an open sentence's truth depends on variables or conditions. Example: Statement: '5 > 3' (always true). Open sentence: 'x + 4 = 8' (true only whe…

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Conjunction connects two statements with 'and'. p ∧ q is true only when both p and q are true. Truth Table: p | q | p ∧ q T | T | T T | F | F F | T | F F | F | F Example: '2 is even AND 3 is odd' (T…

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Disjunction connects two statements with 'or'. p ∨ q is false only when both p and q are false (inclusive or). Truth Table: p | q | p ∨ q T | T | T T | F | T F | T | T F | F | F Example: '2 is even …

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Negation reverses the truth value of a statement using 'not'. Truth Table: p | ~p T | F F | T Example: p: '5 is odd' (True), ~p: '5 is not odd' (False) Note: ~(~p) ≡ p (double negation law)…

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Conditional statement 'If p then q' (p → q) is false only when p is true and q is false. Truth Table: p | q | p → q T | T | T T | F | F F | T | T F | F | T Example: 'If it rains, then roads are wet'…

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Biconditional 'p if and only if q' (p ↔ q) is true when both statements have the same truth value. Truth Table: p | q | p ↔ q T | T | T T | F | F F | T | F F | F | T Example: 'A triangle is equilate…

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Step-by-step solution: Given: p = T, q = F, r = T Step 1: Calculate p ∧ q = T ∧ F = F Step 2: Calculate p ∧ r = T ∧ T = T Step 3: Calculate (p ∧ q) ∨ (p ∧ r) = F ∨ T = T Answer: True (T)…