Question

If the universe of discourse is the set of real numbers, what are the truth values of the following?

1. ∀x (x 2 6= x)
2. ∃x (x 2 > x)
3. ∃x (x 2 = 2)
4. ∃x (x 2 = −1)
5. ∃x (x 2 +2 > 1)

Answer 1

The truth values of given propositions within the realm of real numbers include: a false statement for the first, and true for the second, third, and fifth; while the fourth is false as it contemplates a square root of a negative number, which is not possible in the set of real numbers.

Step-by-step explanation:

The universe of discourse being the set of real numbers pertains to mathematical propositions where variables represent real numbers. In this context, examining the truth values for several mathematical statements yields the following:

1. ∀x (x^2 ≠ x) states that for all x, x squared is not equal to x. This statement is false because, for instance, for x = 1 or x = 0, x squared is indeed equal to x.
2. ∃x (x^2 > x) asserts that there exists an x such that x squared is greater than x. This statement is true as any real number greater than 1 or less than -1 satisfies this condition.
3. ∃x (x^2 = 2) suggests the existence of an x such that x squared equals 2. This is true because √2 is a real number, and (√2)^2 = 2.
4. ∃x (x^2 = −1) implies that there is a real number x whose square is -1. This statement is false, as the square of any real number cannot be negative.
5. ∃x (x^2 + 2 > 1) indicates there is an x such that x squared plus 2 is greater than 1. This is true, considering any real number will make the inequality correct. For example, if x = 0 then 0^2 + 2 = 2 which is greater than 1.