Answer to Question #94844 in Discrete Mathematics for Amrit

Question

If the domain of discourse is all integers, find a counterexample*, if possible, to the following universally quantified statements:
a. ∀x∃y(x = 1/y)
b. ∀x∃y(y2 −x < 100)
c. ∀x∀y(x2= y3)

Answer 1

a). Counterexample x=0, in this case there is no y since 0*y must be 1, but in fact, 0*y is equal to 0.

b). Counterexample is any integer x which is less than -100 (e.g. -101, -102, ...)

In this case there is no y because the square of y cannot be less than 0.

c). Counterexample - for instance, x=1 and y=2, the square of x is 1, while the cube of y is 8. It is obvious they are not equal.