# cardinality of a infinite subset

#### lamsung

##### New member
I saw the below statement which is intuitively correct:

If a set has cardinality m then none of its subsets has cardinality greater than m.

Is it necessarily true for a infinite set case?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
If a set has cardinality m then none of its subsets has cardinality greater than m.

Is it necessarily true for a infinite set case?
Of course. If a subset $B$ of $A$ has cardinality strictly greater than the cardinality of $A$ itself, then there is an injection from $A$ to $B$, but not from $B$ to $A$, by the Cantor–Bernstein–Schroeder theorem. For an infinite setm, it is possible to have an injection into a proper subset, but there is also a trivial injection (inclusion) from a subset to the whole set.

If you need more details, tell us what $m$ is here and what is the definition in your context of having cardinality $m$ or greater than $m$.