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cardinality of a infinite subset

lamsung

New member
Jul 19, 2013
5
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
Jan 30, 2012
2,492
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 $m$.