Proving a Set

Printable View

May 9th 2006, 07:01 PM
PhilipJ

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.
May 9th 2006, 07:15 PM
ThePerfectHacker

Quote:

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.

$(A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$

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

If not then, $\exists x$ such as $x\in A,B,C$
Notice that $x\in ((B\cap (A\cup C))$ because $x\in B$ and $x\in A\cup C$ .
Thus,
$x\in ((B\cap (A\cup C))\cup (C \cap A))$
Thus, any element of $(A\cap B\cap C)$ is element of $((B\cap (A\cup C))\cup (C \cap A))$ . Thus, by definition,
$(A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$
May 9th 2006, 07:24 PM
ThePerfectHacker

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

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

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

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

Thus, (by transitive propetry),
$<br /> (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))<br />$