I just don't seem to get proofs or set theory so hopefully my question makes sense.
I'm not sure when I should or shouldn't use an example to prove or disprove a statement?
One example question is, if C $\subseteq$ A and D $\subseteq$ B, then C $\cup$ D $\subseteq$ A $\cup$ B.
I want start by making set A = $\{1, 2, 3, 4, 5\}$ , B = $\{1, 2, 3, 4, 5, 6\}$ C = $\{1, 2, 3\}$ and D = $\{1,2,3,4\}$
and this would show an example proving this statement. However I think this might be wrong because it only shows one example.
So, I tried to think of a counterexample that would show that the statement is false. However I'm not sure if I should then try to prove, If C $\subseteq$ A and D $\subseteq$ B, then C $\cup$ D $\subseteq$ A $\cup$ B false or if I should be proving, If C $\subseteq$ A and D $\subseteq$ B, then C $\cup$ D $\subsetneq$ A $\cup$ B false??
I've also tried using x $\in$ C $\subseteq$ A, then x $\in$ C and x $\in$ A, x $\in$ D
$\subseteq$ B, then x $\in$ D and x $\in$ B
But, I didn't know what to do from there.
Best Answer
In this particular case, you are asked to prove a "for all" type question. For all sets $A, B, C$, and $D$ with $C\subseteq A$ and $D\subseteq B$ prove that $C\cup D\subseteq A\cup B$.
To show the containment, take an arbitrary element from the left and show that it is in the right. For example, let $x\in C\cup D$, then either $x\in C$ or $x\in D$. Without loss of generality, suppose that $x\in C$. Then $x\in A$ since $C\subseteq A$ and hence $x\in A\cup B$.