midterm2Cribsheet - Equations of Boolean Algebra page 1...

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Equations of Boolean Algebra page 1 From Fig 2.1, Hal & O’Donnel, Discrete Math with a Computer , Springer, 2000 Additional equations: (a b) b = b { absorption } (a b) b = b { absorption } (a b) c = (a c) (b c) { imp } page 2 (( x.f(x)) q) = (( x.(f(x) q)) { dist over } (( x.f(x)) q) = (( x.(f(x) q)) { dist over } (( x.f(x)) q) = (( x.(f(x) q)) { dist over } (( x.f(x)) q) = (( x.(f(x) q)) { dist over } Equations of Predicate Calculus ( x. f(x)) = ( y. f(y)) { R} ( x. f(x)) = ( y. f(y)) { R} ( x. f(x)) f(c) {7.3} f(c) ( x. f(x)) {7.4} ( x. ¬ f(x)) = ( ¬ ( x. f(x))) {deM } ( x. ¬ f(x)) = ( ¬ ( x. f(x))) {deM } x n o t f r e i q ( x.(f(x) g(x))) = (( x.f(x)) ( x.g(x))) { dist over } (( x.f(x)) ( x.g(x))) ( x.(f(x) g(x))) {7.12} (( x.f(x)) ( x.g(x))) ( x.(f(x) g(x))) {7.13} ( x.(f(x) g(x))) = (( x.f(x)) ( x.g(x))) { dist over } y not free in f(x) and x not free in f(y) Principle of Mathematical Induction another way to skin a cat ± { I} — an inference rule with n. P(n) as it’s conclusion P(0) n.P(n) P(n+1) ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ {Ind} n. P(n) Induction ± One way to use { I} ² Prove P(0) ² Prove P(n +1) for arbitrary n 9 Takes care of P(1), P(2), P(3), … ± Mathematical induction makes it easier ² Proof of P(n +1) can cite P(n) as a reason 9 If you cite P(n) as a reason in proof of P(n+1), your proof relies on mathematical induction 9
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midterm2Cribsheet - Equations of Boolean Algebra page 1...

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