# Thread: Representation of set. Cartesian product

1. ## Representation of set. Cartesian product

Hello.
When we say A*B={(x,y) : x is an element of A, y is an element of B}, we know that:
if (x,y) is an element of A*B => x is an element of A, y is an element of B

Is it also true?
If x is an element of A, y is an element of B => (x,y) is an element of A*B

Thanks.

2. Originally Posted by Researcher
Hello.
When we say A*B={(x,y) : x is an element of A, y is an element of B}, we know that:
if (x,y) is an element of A*B => x is an element of A, y is an element of B

Is it also true?
If x is an element of A, y is an element of B => (x,y) is an element of A*B

Thanks.
Yes, by definition. $\displaystyle (x,y) \in A \times B$ if $\displaystyle x \in A, y \in B$. But as that is a definition the "if" can be read as an "if and only if". The "only if" gives you the direction you need.

3. Thanks Swlabr.
Could you please introduce a source containing it to me?

4. Originally Posted by Researcher
Thanks Swlabr.
Could you please introduce a source containing it to me?
Any book on elementary set theory should have it, although I can't give you specific example.