Set theory questions - Subsets from Zorich Mathematical Analysis I

Question

I am doing a text that my big brother gave me: Mathematical Analysis I - Zorich. This stuff is pretty hard for me, since in class we don't do sets.

I can see why they are true with pictures, but i don't know how to prove them mathematically:

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$A,B,C\subset M$

1.a) $(A \subset C) \wedge (B \subset C) \Leftrightarrow ((A \cup B) \subset C)$

I can see that if the left is true, the right is true, and if one is false, the other is false.

but what if $A$ exclusively OR $B$ subsets C, then the right is true, but the left is false?

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1.b) $(C\subset A) \wedge (C \subset B) \Leftrightarrow (C\subset (A \cap B))$

This seems obvious, but how to mathematically show it? i can just draw the Venn diagram again, but that is not math?

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Side question: Do Venn diagrams count as mathematical proofs? Or are they just a tool to clear things up?

Answer 1

If $A$ exclusively or $B$ are contained in $C$, then $A\cup B$ is not a subset of $C$. Without loss of generality, we can assume that $A\subset C$ and $B\not\subset C$. If $A\cup B\subset C$ then $B\subset C$.

I'll give the hint of 1.b): use the definition of intersection $$A\cap B=\{x:x\in A\land x\in B\}.$$

Let assume $x\in C$ then $x\in A$ and $x\in B$ so...

Answer 2

$$(C\subset A) \wedge (C \subset B) \Leftrightarrow (C\subset (A \cap B))$$

Proof: $$(C\subset A) \wedge (C \subset B) \implies (C\subset (A \cap B))$$ Assume $C\subset A$ and $C\subset B$. This means $\forall x\in C, x\in A$ and $x\in B$. From this we can see that $\forall x\in C,x\in (A\cap B)$. Since this is all $x\in C$, $C\subset A\cap B$.

We must now prove that:$$(C\subset A) \wedge (C \subset B) \Longleftarrow (C\subset (A \cap B))$$

Assume $C\subset (A\cap B)$ hence $\forall x\in C, x\in A\cap B$, since $(A\cap B) \subset A$ and $(A\cap B) \subset B$, $\forall x\in C,x\in A$ and $x\in B$

thus it is proved.