If and only if there is a bijection from a set onto itself, then the set is of distinct terms?

Question

Is it true that iff CardA = Card A then A is a set of distinct terms?

[This questions is actually from a confusion on what a set versus multiset is]

Answer 1

If $A$ is a set it is defined to have no duplicate elements. Because Card $A$ = Card $A$ is a tautology, your statement is of the form True iff $A$ is a set of distinct terms. As long as $A$ is a set this will be true.

Answer 2

I assume your question is:

Is it true that for any set $A$: $|A|=|A|$ if and only if $A$ is a set of distinct elements?

Well, sets never have repeating elements. So, for sets, both sides of the if and only if are true for any set: for any set $A$: $|A|=|A|$, and $A$ does not have repeating elements. So, the statement is true since both sides of the if and only if are true for any set $A$.

But, if we do allow sets to have repeating elements, i.e. if this question is really about multisets, rather than sets, then the statement is false: It is still true for any multiset $A$ that $|A|=|A|$, but clearly a multiset $A$ can contain repeating elements.