Prove: if $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$

It's quite obvious, but I'm not sure what the proper approach is to proving a set problem that involves subsets, as none of the set identities given in our textbook include subsets.


Usually what you want to do is pick any arbitrary element of $A \cup B$, call it $x$. So $x \in A \cup B$. Then show that $x \in C$, using what you know about elements of $A$ and $B$.


To show $A\cup B \subseteq C$, we show that for any $x \in A\cup B$, it follows that $x \in C$.

Assumptions (Givens): $A\subseteq C$, $B \subseteq C$.

Suppose $x \in A\cup B.\;\;$ Then $x \in A$, or $x\in B\;$ (from the definition of set union).

Now, use what you know from our givens above to argue, therefore, $x \in C$.

Thus, we will have shown $A\cup B \subseteq C$.


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