Thread: 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.

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.

If there is nothing to prove because the empty set is subset of every set.

If not then, such as
Notice that because and .
Thus,

Thus, any element of is element of . Thus, by definition,

Also, using the "transitive" property of subsets:
and then,

And the property,


And finally the property,

---------------
Thus,

Because

And,

Because

Thus, (by transitive propetry),