# a "sets" question.

• Oct 13th 2008, 03:42 PM
kalle
a "sets" question.
Hello to everyone!
Could someone say me the answer of the following set question?

A ∩ B ⊆ A ∪ B = Prove!

It needs a proving like the example below:

A ∩ B = {x|x€A and x€B}

Thanks,
Best wishes.
• Oct 13th 2008, 08:43 PM
Jhevon
Quote:

Originally Posted by kalle
Hello to everyone!
Could someone say me the answer of the following set question?

A ∩ B ⊆ A ∪ B = Prove!

It needs a proving like the example below:

A ∩ B = {x|x€A and x€B}

Thanks,
Best wishes.

Lets consider the case where both sets are non-empty. the result holds trivially in the cases where both are empty or $\displaystyle A \cap B$ is empty. (why?)

to prove $\displaystyle A \cap B \subseteq A \cup B$ we must show that if $\displaystyle x \in A \cap B$, then $\displaystyle x \in A \cup B$.

Assume $\displaystyle x \in A \cap B$. Then $\displaystyle x \in A$ and $\displaystyle x \in B$. Clearly it holds that $\displaystyle x \in A$ or $\displaystyle x \in B$, so that $\displaystyle x \in A \cup B$. Thus, $\displaystyle A \cap B \subseteq A \cup B$