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

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The discussion centers on the validity of two mathematical statements regarding set inclusion. The first statement, asserting that if A is a subset of both B and C, then A is a subset of their union, is true in one direction but false in the reverse. The second statement, claiming that if A is a subset of either B or C, then A is a subset of their intersection, is also false in one direction but true in the other. Counterexamples are suggested to illustrate the inaccuracies of both statements. Understanding these nuances is crucial for grasping set theory concepts.
cbarker1
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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
 
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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 $C=\emptyset$.
    -
  2. This time $\Leftarrow$ is true but not $\Rightarrow$. Consider $A\subset B$, $A\ne\emptyset$, and $B\cap C=\emptyset$.
 

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