1. ## Proving a Set

Note: ^ = "intersects"; U = "union"

Prove that A ^ B ^ C "is a subset of" D, where D = (B ^ (A U C)) U (C ^ A)

Obviously, when drawing a Venn diagram, they all will meet at the very middle. How can I prove this is true, in a formal proof.

2. Originally Posted by PhilipJ
Note: ^ = "intersects"; U = "union"

Prove that A ^ B ^ C "is a subset of" D, where D = (B ^ (A U C)) U (C ^ A)

Obviously, when drawing a Venn diagram, they all will meet at the very middle. How can I prove this is true, in a formal proof.
$\displaystyle (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$

If $\displaystyle A\cap B\cap C=\{\}$ there is nothing to prove because the empty set is subset of every set.

If not then, $\displaystyle \exists x$ such as $\displaystyle x\in A,B,C$
Notice that $\displaystyle x\in ((B\cap (A\cup C))$ because $\displaystyle x\in B$ and $\displaystyle x\in A\cup C$.
Thus,
$\displaystyle x\in ((B\cap (A\cup C))\cup (C \cap A))$
Thus, any element of $\displaystyle (A\cap B\cap C)$ is element of $\displaystyle ((B\cap (A\cup C))\cup (C \cap A))$. Thus, by definition,
$\displaystyle (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$

3. Also, using the "transitive" property of subsets:
$\displaystyle X\subseteq Y$ and $\displaystyle Y \subseteq Z$ then, $\displaystyle X\subseteq Z$

And the property,
$\displaystyle X\cap Y\subseteq X$

And finally the property,
$\displaystyle X\subseteq X\cup Y$
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Thus,
$\displaystyle B\cap (A\cap C)\subseteq A\cap C$
Because $\displaystyle X=B,Y=A\cap C$

And,
$\displaystyle A\cap C\subseteq ((B\cap (A\cup C))\cup (C \cap A))$
Because $\displaystyle X=(C \cap A)),Y=((B\cap (A\cup C))$

Thus, (by transitive propetry),
$\displaystyle (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$