Question

8. (10 pts.) Find a common domain for the variables x, y, and z for which the statement is true and another domain for which it is false. Explain your answer.

Answer 1

Given statement,

∀x ∀y[ (x ≠ y) $\rightarrow$ ∀z((z=x)V(z=y))]       ------------------(i)

The above statement is equivalent to,

∀x ∀y∀z [ (x ≠ y) $\rightarrow$ (z=x)V(z=y)]

which is again equivalent to,

∀x ∀y∀z [ ((z≠x)^(z≠y)) $\rightarrow$ (x=y)]          --------(ii)

The statement (i) states that "for all x, for all y, for all z, if x is not equal to y then z is either x or y."

The statement (ii) states that "for all x, for all y, for all z, if z is not equal to x and z is not equal to y then x and y are equal."

According to the given statement we can conclude that the given statement will be true only if " there exists exactly two different numbers in the given domain."