Proving A Subset B & C $\Rightarrow$ A Subset (B $\cup$ C)

[cbarker1]

Dear Everybody,

I am struggling now for determining if the following statements are true or false. If the statement is true, then prove it. If not, make a counterexample.
Here are the statements:

1. $A \subset B$ and $A \subset C$ if and only if $A \subset (B\cup C)$.
2. $A \subset B$ or $A \subset C$ if and only if $A \subset (B\cap C)$.

My attemption:
Let A={1,2,3}, B={1,2,4}, and C={3}.

1. I believe this is true. $A\subset B$ and $A\subset C$. Thus $A \subset (B\cup C)$ is true.
2. I don't know how to begin.

Thanks,
Cbarker1

 

[Olinguito]

Hi Cbarker1.

The two statements are in fact false, thus you just need to find a counterexample to disprove each. Hints:

1. The $\Rightarrow$ part is true but not the $\Leftarrow$ part. Consider $A\subset B$, $A\ne\emptyset$, and $C=\emptyset$.
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2. This time $\Leftarrow$ is true but not $\Rightarrow$. Consider $A\subset B$, $A\ne\emptyset$, and $B\cap C=\emptyset$.