Answer to Question #261608 in Discrete Mathematics for Detactive

Question

Find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which it is false

Answer 1

Let us find a common domain for the variables $x, y, z,$ and $w$ for which the statement $∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z))$ is true and another common domain for these variables for which it is false.

Let the domain $D$ contain four elements, $D=\{1,2,3,4\}.$ Then for each elements $x,y,z\in D$ the set $D\setminus \{x,y,z\}$ contains at least one element, and hence we can get $w\in D\setminus \{x,y,z\}.$

Therefore, for this domain the statement $∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z))$ is true.

Let the domain $D'$ contain one elements, $D'=\{1\}.$ Then for each elements $x,y,z,w\in D'$ it follows that $x=y=z=w=1,$ and hence there is no element $w\in D'$ such that $(w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z).$

Therefore, for this domain $D'$ the statement $∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z))$ is false.