Question

Find the truth set of each predicate. (If your answer is an interval, enter it using interval notation; otherwise enter it as a comma-separated list.) (a) Predicate: 10 d is an integer, domain: ℤ Correct: Your answer is correct. (b) Predicate: 10 d is an integer, domain: ℤ+ Correct: Your answer is correct. (c) Predicate: 1 ≤ x2 ≤ 4 , domain: ℝ (d) Predicate: 1 ≤ x2 ≤ 4 , domain: ℤ Correct: Your answer is correct.

Answer 1

(a) Predicate: 10d is an integer

The domain is $\mathbb{Z}$. In fact, integers are closed under multiplication, and 10 is an integer. So, if d is an integer, 10d is a multiplication between integers, and thus an integer.

(c) $1\leq x^2\leq 4$

Since 1 and 4 are both squares, we have

$1\leq x^2 \leq 4 \iff 1^2\leq x^2 \leq 2^2$

And we deduce that x must be between 1 and 2. But since negative numbers become positive when squared, the numbers between -2 and -1 are also fine. So, the domain is

$[-2,-1]\cup [1,2]$

Predicates (b) and (d) are identical to (a) and (c)