# Cardinality of Cartesian Product

sujoykroy
Can you prove the following theory of cardinality for a Cartesian product, -
$\left|\:A\:\right|\:\leq\:\left|\:A\:\times\:B\:\right|\: if\: B\neq\phi$

In English,
The cardinality of a set $A$ is less than or equal to the cardinality of Cartesian product of A and a non empty set $B$.

Staff Emeritus
Gold Member
What have you tried? What methods can you use? What ways of restating the problem have you considered?

Homework Helper
Hi sujoykroy! In problems like this, just write out the definition, and then plug the problem into it.

So … what is the definition of "cardinality of P ≤ cardinality of Q"? oh … and … what is the definition of "non empty set"? sujoykroy
What have you tried? What methods can you use? What ways of restating the problem have you

considered?

I think, if you pick up a binary relation $f$ in such way that $f\left(\:a\:\right)\:=\:\left(\:a\:,\:b\:)$ for some $b\:\in\:B$ for all $a\:\in\:A$, then $f$ will be a one-to-one function with $dom\:f\:=\:A$ and $ran\:f\:\subset\:A\:\times\:B$, hence proving that $\left|\:A\:\right|\:\leq\:\left|\:A\:\times\:B\:\right|\: if\: B\neq\phi$, but i am not sure if the approach is right or not.

Hi sujoykroy! In problems like this, just write out the definition, and then plug the problem into it.

So … what is the definition of "cardinality of P ≤ cardinality of Q"? oh … and … what is the definition of "non empty set"? Below is the definition of cardinality that i am using,
The cardinality of a set $A$ is less than or equal to the cardinality of a set $B$ if there is a one-to-one function $f$ on $A$ into $B$

Staff Emeritus
Gold Member
but i am not sure if the approach is right or not.
Well, try formalizing it. If you wind up with a valid proof, then your approach is right. sujoykroy
Well, try formalizing it. If you wind up with a valid proof, then your approach is right. Thanks. Actually i was trying to understand/prove the use/existence of Infinite Sequence used in various proof of Cantor-Schroder-Bernstein i.e. if $\left|X\right|\:\leq\:\left|Y\right|$ and $\left|Y\right|\:\leq\:\left|X\right|$ then $\left|X\right|\:=\:\left|Y\right|$ and current problem was a doorway to open up the logical window towards it. So, formalization was not really my problem, i just needed to get confirmation if the logic is correct.

Homework Helper
Below is the definition of cardinality that i am using,

"The cardinality of a set A is less than or equal to the cardinality of a set B if there is a one-to-one function f on A into B"

Hi sujoykroy! Yes, that's the one … so, in this case, you need to define a one-to-one f on A into A x B.

And to do that, answer the question: what is the definition of "non empty set"?

(it may sound a daft question … but sometimes maths is like that! )

rich292
I also have a quick query regarding something related to cardinality of a cartesian product.

What does $$\left|A\right|$$ = $$\left|A \times \aleph\right|$$for any set A, tell you about A?

I hope to use this to find an injective function from $$\aleph^{A}$$to $$\left\{0,1\right\}^{A}$$