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

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(, y) differently for each part. To define the predicate, you can either use known predicates on numbers (e.g. " and y are both even" or "r > y", etc.) or by defining explicitly for which x, 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 3VyP(x, y) are not logically equivalent. Justify your answer. (b) Give a counterexample which proves that are not logically equivalent. Justify your answer. (c) Give a counterexample which proves that are not logically equivalent. Justify your answer.

Explanation / Answer

a) let x is student and y is book then ThereExists x for all y p(x,y) implies that All books belongs to students.

Now ThereExists y for all x P(x,y) implies that All students have book.

Which is logically not equivalent

b)let x be the humans and y be a pen then For all x ThereExists y P(x,y) implies that All humans has pen

Whereas ThereExists y For all x P(x,y) implies that every pen belongs to all human which is logically not equivalent

c)let x be the animals and y be the elephant such that For all x ThereExists y P(x,y) implies that Elephant is an animal

Whereas ThereExists x for all y P(x,y) implies that Animal is one of the Elephant which is logically not equivalent.hence proved

  

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