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Logic Summary: Symbols, formulas, truth tables A truth table involving n different statements will have 2n rows. Examples: The truth table for (~p V q) V r has 23=8 rows and the truth table for p V (q → ~q) has 22 rows. Connective and or not If …then If and only if Symbol Λ V ~ → ↔ Name Conjunction Disjunction Negation Implication Biconditional The 5 BASIC truth tables: Negation, 'not p' p ~p T F F T Conjunction, 'p and q' "T to F, F to T." Disjunction, 'p or q' p q pΛq p q pVq T T F F T F T F T F F F T T F F T F T F T T T F "Only True if both are True." "Only False if both are False." Implication, 'If p, then q' Biconditional, 'p if and only if q' p q p→q p q p↔q T T F F T F T F T F T T T T F F T F T F T F F T "Only False if p is True and q is False." DeMorgan’s Laws ~(p Λ q) ≡ ~p V ~ q ~(p V q) ≡ ~p Λ ~ q "Only True if p and q have the same truth value." Statement: p→q (If p, then q): If you go, then I stay. Converse: q→p (If q, then p): If I stay, then you go. Inverse: ~p→~q (If not p, then not q): If you do not go, then I will not stay. Contrapositive: ~q→~p (If not q, then not p): If I do not stay, then you do not go. A statement p → q is equivalent to its contrapositive ~q → ~p. For p → q, p is the ‘hypothesis’ and q is the ‘conclusion’. Quantifiers: Universal (All, Each, Every, No, None) and Existential (Some, There exists, At least one) NEGATION OF A STATEMENT WITH A UNIVERSAL QUANTIFIER WILL INVOVLE AN EXISTENTIAL QUANTIFIER. NEGATION OF A STATEMENT WITH AN EXISTENTIAL QUANTIFIER WILL INVOVLE A UNIVERSAL QUANTIFIER. (Some tips on negating statements with quantifiers are on the following page.) Two statements are logically equivalent if their truth tables are identical Basic circuit (with a gate that can open and close) Circuit "in series" (Gates lined up one after the other) -'Current' will only flow all the way across if both gates are closed. -If 'open' is False & 'closed' is True, this circuit is logically equivalent to the CONJUNCTION ( p Λ q ). Circuit "in parallel" (Gates are arranged one above the other) -'Current' will not flow all the way across only if both gates are open. -If we define 'open' as False & 'closed' as True, this circuit is logically equivalent to the DISJUNCTION ( p V q ). Here are a few tips for negating statement with quantifiers. The negation of an "All do/are" statement ["All athletes are wealthy."] is a "Some do not/are not" statement ["Some athletes are not wealthy."] or a "There exists at least one not" statement ["There exists at least one athlete who is not wealthy."] The negation of an "All do not/are not" statement ["It is true for all athletes that they are not wealthy."] is a "Some do/are" statement ["Some athletes are wealthy."] or a "There exists at least one" statement ["There exists at least one athlete who is wealthy."] The negation of a "Some do/are" statement ["Some students get scholarships."] is a "No/none do" statement ["No student gets a scholarship."] or a "All do not/are not" statement ["It is true for all students that they do not get scholarships."] The negation of a "Some do not/are not" statement ["Some students do not get scholarships."] is a "No/none do not" statement ["No student does not get a scholarship."] or a "All do/are" statement ["All students get scholarships."]