4 (9 points) In each part of this question, build your counterexample with unive

Question

4 (9 points) In each part of this question, build your counterexample with universe (domain) -1,0,1) You can define the predicate P(,) differently for each part. To define the predicate, you can either use known predicates on numbers (e.g. "z and y are both even" or "x > y", etc.) or by defining explicitly for which r, y values P(x, y) evaluates to T and for which values it evaluates to F (to do this, you must consider all possible domain values). (a) Give a counterexample which proves that 3rVyP(r,u) ByvrPr, y) are not logically equivalent. Justify your answer (b) Give a counterexample which proves that Vay P(r,y) are not logically equivalent. Justify your answer (c) Give a counterexample which proves that BrVyP, are not logically equivalent. Justify your answer.

Explanation / Answer

Hi,
1. Let the predicate be x+2y is even.
now case 1: for all y, there exists x such that x+2y is eveb, this is true, if x is even, x+2y becomes even
case2: for all x,there exists y such that x+2y is even, thisi is false, because if x is odd, no value of y can make x+2y even, hence they are not logically equivalent
2.Let the predicate be y+2x is even.
now case 1: for all x, there exists y such that y+2x is eveb, this is true, if y is even, y+2x becomes even
case2: there exists y,for all x such that 2x+y is even, thisi is false, because if y is odd, no value of y can make 2x+y even, hence they are not logically equivalent

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