What is the union of the set $\mathbb Z$ and $\mathbb Z$?

Question

Exercises 1.20 (ii) asked in, How to Think Like a Mathematician by Kevin Houston.

My solution:

$$\{x\ |\ x ∈ \mathbb Z\}$$

Is this correct?

Answer 1

• For any set $A$, we have $A \cup A = A$

• $\{ x|x \in \mathbb{Z}\}=\mathbb{Z}$

Answer 2

Yes you are correct.

In the answer above, we still need to justify why for any set $A$, $A \cup A = A$. The set theory approach is that the following:

For any set $X, Y$, $X \subseteq Y$ and $Y \subseteq X$ if and only if $X = Y$.

So we need to show that $A \cup A \subseteq A$ and $A \subseteq A \cup A$. For any $a \in A \cup A$, $a \in A$ or $a \in A$ (sounds redundant and redundant). Hence $a \in A$. Then $A \subseteq A \cup A$ is obvious. Hence $A \cup A = A$.

Now take $A$ to be $\mathbb{Z}$.