predicate Calculus Solution Outline

predicate Calculus Solution Outline - by F1. Proved....

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Solution to Pred. Calculus Problems. Note: V stands for "forall" E stands for "there exists" Problem 1. a. Vx (P(x) => Q(X)) b. Ex (Q(x) => P(X)) c. not (Vx (Q(x) => P(X))) Problem 3. Denote St. Francis by constant sf Denote x loves y by l(x,y) where x and y are variables F1: St. Francis is loved by every one who loves some one: === Vx Ey (l(x,y) => l(x, sf) F2: Noone loves nobody (which means every one loves someone) === Vx Ey l(x,y) Deduce: Vx l(x,sf) Every one loves someone by F2, every one must also love St. Francis
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Unformatted text preview: by F1. Proved. Problem 3. o Vx Vy Ez P(x,y,z) o Vx Vy Vz Vx ((P(x,y,z) and P(x,y,w)) => E(z,w)) o Vx Vy Vs Vz Va Vt Vb ((P(x,y,s) and P(s,z,a) and P(y,z,t) and P(x,t,b)) => E(a,b) Problem 4. o P(a, f(a)) and P(b, f(b)) = P(1, f(1)) and P(2, f(2)) = P(1, 2) and P(2, 1) = T and F = F o Vx Ex P(y,x) Consider y=1 then for x = 1, P(y, x) = P(1,1) = true Consider y=2 then there does not exist x such that P(2, x) = true So Vx Ex P(y,x) is false under this interpretation o left as an exercise...
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predicate Calculus Solution Outline - by F1. Proved....

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