Transcription of Logic, Sets, and Proofs - Amherst
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Logic, Sets, and Proofs - Amherst
Logic, Sets, and ProofsDavid A. Cox and Catherine C. McGeochAmherst College1 LogicLogical statementis a mathematical statement that can beassigned a value eithertrueorfalse. Here we denote logical statements with capitallettersA,B. Logical statements be combined with the following operators to formnew logical nameNotation I Notation II JavaAND (Conjunction)A BA BA&&BOR (Disjunction)A BA+BA||BNOT (Negation) A A!AIMPLIES (Implication)A BifAthenBIF AND ONLY IF (Equivalence)A BAiffB== is a list of tautologies. In any proof, you can replace a statementin the first column with the corresponding statement in the second column, and viceversa. All of these can be proved by truth statement DescriptionA BB A is commutativeA BB A is commutative(A B) CA (B C) is associative(A B) CA (B C) is associativeA (B C) (A B) (A C) distributes over A (B C) (A B) (A C) distributes over A falseAfalse is identity for A trueAtrue is identity for A Atruelaw of excluded middleA Afalsecontradict
Exam-ples: “x is even” and “x > y” are predicates. The truth of the predicate depends on which particular members of the universe are plugged in for the variables. We combine quantifiers with predicates to form statements about members of U. There are two basic types: 3
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