Question

The notation

∃!xP(X)
denotes the proposition
"There exists a unique x such that P(x) is true."
If the universe of discourse is the set of integers, what are the truth values of the following?
1. ∃!x(x = x + 1)
2. ∃! x(x² = 1)
3. ∃!x(x + 3 = 2x)
4. ∃!x(x > 1)

Answer 1

The truth values of the given statements are false, false, true, and false respectively, based on the criterion of uniqueness for the proposition ∃!xP(x).

Step-by-step explanation:

Given the notation ∃!xP(x) which denotes the proposition 'There exists a unique x such that P(x) is true,' we can now evaluate the truth values of the following statements:

1. ∃!x(x = x + 1): This statement claims the existence of a unique integer x such that x is equal to x + 1. However, no such integer exists. For any value of x, x will never be equal to x + 1. Therefore, the statement is false.
2. ∃!x(x² = 1): This statement claims the existence of a unique integer x such that x squared is equal to 1. The only integers that satisfy this condition are x = 1 and x = -1. Since there are two possible values of x that make the statement true, it is false according to the notation's requirement of uniqueness.
3. ∃!x(x + 3 = 2x): This statement claims the existence of a unique integer x such that x + 3 is equal to 2x. Simplifying the equation, we get 3 = x, which means x must equal 3 for the statement to be true. Since x = 3 is the only solution, the statement is true.
4. ∃!x(x > 1): This statement claims the existence of a unique integer x greater than 1. Since there are multiple integers greater than 1, such as 2, 3, 4, and so on, the statement is false as it fails to satisfy the uniqueness requirement.