Question

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. Sets \(\mathbb{Z}\) and \(\mathbb{N}\) are infinite.

Short answer

The statement 'Sets \(\mathbb{Z}\) and \(\mathbb{N}\) are infinite' is a true statement.

Step-by-step solution

Understanding the given sets

Set \(\mathbb{Z}\) includes all integers, meaning it includes all negative numbers, zero, and all positive numbers. This set has no end in either the positive or negative direction - it can go on indefinitely. So, \(\mathbb{Z}\) is an infinite set. Set \(\mathbb{N}\) includes all natural numbers, which are positive integers including zero. This set has no end in the positive direction - it can increase indefinitely. So, \(\mathbb{N}\) is also an infinite set.

Deciding if the statements are true

Since we have determined that both \(\mathbb{Z}\) and \(\mathbb{N}\) are infinite, the statement 'Sets \(\mathbb{Z}\) and \(\mathbb{N}\) are infinite' is truthfully expressing a fact. Therefore, this statement is true.