MATH221 Lecture Notes - Lecture 2: Logical Equivalence, Contraposition, Double Negative
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~ (p q) (~ p) (~ q) *don"t skip steps for full marks and understanding. Given: ( (p q) r ) ( (~ p r) (q r) ) Let (*) be the tautology r s ~ r s ( (p q) r) (~p q) r by (*) ( ~ (~ p q) r ) by (*) (~ ~ p ~ q) r demorgan"s. (p ~q) r double negative. (p r) (~ q r) distributive. (~ (~ p) r) (~ q r) double negative. Notation: universal qualifier: symbol denotes for all . Belongs to / element of ; i. e. 3 . The negation of all are" is some is" / "are not" i. e. all real numbers have a positive or zero square . = x , x2 0 existential quantifier: symbol denotes there exists i. e. there exists integers m, n satisfying m + n = mn . = m, n , m, n .
