1. Let P(x,y) be the predicate x-y = 20 where x and y are integer numbers. For each of the following statement indicate whether or not they true and justify your answer.

   1. (x) (y), P(x,y)

   2. (x) (y), P(x,y)

2. What is the negation of the following proposition?

   integers a, b and c, if a-b is even and b-c is even, then a-c is even

3. Rewrite each of the following proposition using quantifiers, predicates and variables. Make sure that you clearly define the predicates, variables and domains.

   1. There exists a program that gives the correct answer to every question that is posed to it.

   2. For all integers m and n, if m*n is even then either m is even or n is even

4. Using De Morgans law for quantified statements and laws of propositional logic prove that:

   ~x (~P(x) Q(x)) x (~P(x) ~Q(x))