Let = p2 , P p1 and + f2 . We use the semantics of rst-order logic without equality. Prove or refute the following formulas semantically. That means you must show that I(A) = T for all models I and assignments (without using a proof calculus) or to give some I, such that

I(A) = F. 1. P(X)

2. X.Y.=(+(X,Y),+(Y,X)) 3. X.(P(X)Y.P(Y))

Problem 9.3 (First-Order Semantics) Let = S, PEE and + E E. We use the semantics of first-order logic without equality. Prove or refute the following formulas semantically. That means you must show that I (A)=T for all models I and assignments (without using a proof calculus) or to give some 1, such that 1 (A) = F. 1. P(X) 2. VX.VY. = (+(X,Y),+(Y, X)) 3. 3X.(P(X) VY.P(Y))