Given that $A$ and $B$ are sets such that $A ⊆ B$, write down $A ∪ B$ in a simplified form.

Question

Since $A$ is a subset of $B$, can I just say that $A ∪ B = B$?

Answer 1

You might need to argue why $A\cup B=B$ is true, but it is. You might argue, why $A\cup B\subseteq B$ and $B\subseteq A\cup B$, which is done in 1-2 sentences. The conclusion gives $A\cup B=B$.

Answer 2

Exercise. Prove
A subset B iff A $\cup$ B = B iff A $\cap$ B = A.

Answer 3

Well, can you?

think it out.

But $A\subset B$ so if $x \in A$ then $x \in B$.

So if $x\in A \cup B$ then either $x \in A$ and therefore in $B$, or $x \in B$. Either way, $x \in B$.

If $x \not \in A \cup B$ then $x$ is in neither $A$ nor $B$. So $x \not \in B$.

So $x\in A\cup B \iff x \in B$.

And that's what it means for two sets to be equal, isn't it? Two sets are equal if they have the same elements. That is to say, if every element is in one set it must be in the other. And when it's not in one it's not in the other.