Cardinality of infinite subset of infinite set

[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

 

[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.

 

[Bipolarity]

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.

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

Thanks!

BiP

 

[Office_Shredder]

The latter. The most common way of seeing the reverse inequality is
http://upload.wikimedia.org/wikiped...ing_natural.svg/429px-Pairing_natural.svg.png

this gives a bijection between $\mathbb{N} \times \mathbb{N}$ and $\mathbb{N}$. It should be clear that you can project $\mathbb{N} \times \mathbb{N}$ onto $\mathbb{Q}$ by (a,b) maps to a/b