Question

4- Find the truth set of each of these predicates where the domain is the set of integers.

a) P(x): ∣x∣=1
b) Q(x): x²=2x
c) R(x): ∣x∣=x
d) S(x): x³ < x²

Answer 1

The truth sets for the predicates P(x), Q(x), R(x), and S(x) are {-1, 1}, {0, 2}, the set of all non-negative integers, and an empty set, respectively.

Step-by-step explanation:

The task is to find the truth set of each given predicate where the domain is the set of integers. Let's examine each predicate one by one.

• P(x): |x| = 1: The truth set for P(x) consists of integers whose absolute value is 1, which are -1 and 1.
• Q(x): x² = 2x: To solve for this, we need to factorize the equation. The truth set for Q(x) comprises integers that solve the equation x(x - 2) = 0, which are 0 and 2.
• R(x): |x| = x: The truth set for R(x) includes all non-negative integers since the absolute value of x equals x only when x is non-negative.
• S(x): x³ < x²: S(x) holds true for integers where the cube is less than the square, which can only occur when 0 < x < 1. However, since we're dealing with integers, the truth set for this predicate is empty.