Question

In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counter example.

(a) ∃x (x + x = 1)
(b) ∃x (x + 2 = 1)
(c) ∀x (x2 − x ≠ 1)
(d) ∀x (x2 − x ≠ 0)

Answer 1

In this problem, the truth of different statements involving integers is evaluated. Some statements are true and some are false, and examples are provided to support the answers.

Step-by-step explanation:

Let's analyze each statement:

(a) ∃x (x + x = 1): This statement is true because there is at least one integer that satisfies the equation, which is 0. Therefore, ∃x (x + x = 1) is true with x = 0.

(b) ∃x (x + 2 = 1): This statement is false because there is no integer that satisfies the equation x + 2 = 1. Therefore, ∃x (x + 2 = 1) is false.

(c) ∀x (x^2 − x ≠ 1): This statement is true because for every integer x, x^2 − x ≠ 1. This can be proven by substituting values and verifying that the equation never holds.

(d) ∀x (x^2 − x ≠ 0): This statement is false because there are integers that satisfy the equation x^2 − x = 0. For example, x = 0 and x = 1 satisfy the equation. Therefore, ∀x (x^2 − x ≠ 0) is false.