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.

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- May 9th 2006, 07:01 PMPhilipJProving 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. - May 9th 2006, 07:15 PMThePerfectHackerQuote:

Originally Posted by**PhilipJ**

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,

$\displaystyle (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$ - May 9th 2006, 07:24 PMThePerfectHacker
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,

$\displaystyle X\subseteq X\cup Y$

---------------

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))

$