# Cardinality of infinite subset of infinite set

1. Dec 6, 2013

### Bipolarity

Am a bit confused about the meaning of cardinality. If $A \subseteq B$, then is it necessarily the case that $|A| \leq |B|$?

I am thinking that since $A \subseteq B$, an injection from A to B exists, hence its cardinality cannot be greater than that of B?

But this cannot be correct, since $\mathbb{Z}$ and $\mathbb{Q}$ have the same cardinality?

Where am I wrong?

Thanks!

BiP

2. Dec 6, 2013

### R136a1

You are right. If $A\subseteq B$, then $|A|\leq |B|$.

In particular, $|\mathbb{Z}|\leq |\mathbb{Q}|$ is true. Don't confuse this with $|\mathbb{Z}|< |\mathbb{Q}|$, which is false.

3. Dec 7, 2013

### Bipolarity

I see. So which of the following is true?
$|\mathbb{Z}|< |\mathbb{Q}|$
$|\mathbb{Z}|= |\mathbb{Q}|$

Thanks!

BiP

4. Dec 7, 2013

### Office_Shredder

Staff Emeritus