Re: cardinals of finite sets

A and B have the same cardinality if there is a 1-1 mapping of A onto B (bijective).

If there is a 1-1 mapping of A onto a subset of B (the same cardinality), there exists a 1-1 mapping into B (injective).

If there is an injective mapping into a subset of B, The mapping is onto that subset from A, and A and that subset have the same cardinaity.

Re: cardinals of finite sets

Quote:

Originally Posted by

**nerazzurri10** Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.

I find it hard to prove it because I can easily find a set A which is:

A={1,2}

B={1,2,3,4}

C={1,2,3}

C is a subset of B. C's cardinality is bigger than A's.

There is an injective function from A to B, of course.

where am I wrong?

If there is an injective function from A to B, it means that A has the same cardinality as *some* subset of B, not *every* subset of B. So, C may have a greater cardinality than A's, but there is another subset of B that has the same cardinality as A's.

Re: cardinals of finite sets

Re: cardinals of finite sets

Nice example of using symbology to reword an existing proof (#2)