Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$

Question

Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$.

MY ATTEMPT

We shall prove that $A\cup B\subseteq C$ iff $A\subseteq C$ and $B\subseteq C$. Let us consider the implication ($\Rightarrow$) first. If $x\in A\cup B$, then $x\in A$ or $x\in B$. In both cases, $x\in C$, once $A\cup B \subseteq C$. Thus we conclude that $x\in C$ implies that $x\in A$ as well as $x\in C$ implies that $x\in B$. Therefore $A\subseteq C$ and $B\subseteq C$, as desired.

Conversely ($\Leftarrow$), if $x\in A\subseteq C$, then $x\in C$. Similarly, if $x\in B\subseteq C$, then $x\in C$. In other words, if $x\in A\cup B$, then $x\in C$, which means that $A\cup B\subseteq C$, as desired.

Could someone check if I am solving it correctly?

Answer 1

The left to right implication is just the first step, try to make the logical structure of the proof more clear:

$\Rightarrow$: We assume the LHS so, $A \subseteq C$ and $B \subseteq C$. Now we want to prove $A \cup B \subseteq C$, so let $x \in A \cup B$. Then $x \in A$ (and then $x \in C$ by $A \subseteq C$) or $x \in B$ (and then $x \in C$ by $A \subseteq C$) and so in either case $x \in C$. Done.

$\Leftarrow$: Now assume $A \cup B \subseteq C$ and we want to show $A \subseteq C$ and $B \subseteq C$. First let $x \in A$. Then $x \in A \cup B$ so $x \in C$ by $A \cup B \subseteq C$. So $A \subseteq C$. Second let $x \in B$, then $x \in A \cup B$ so $x \in C$ again by $A \cup B \subseteq C$. Hence $B \subseteq C$ and both inclusions have been shown.