Logic, Sets, and Proofs - Amherst

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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 AfalsecontradictionA AA is idempotentA AA is idempotent AAdouble negative (A B) A BDe Morgan s law for (A B) A BDe Morgan s law for A B A Brewriting implicationA B B AcontrapositiveA (B C) (A B) Cc

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false. Here we denote logical statements with capital letters A,B. Logical statements be combined with the following operators to form new logical ...

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