Definition:Set Equivalence

Definition
Two sets $$S$$ and $$T$$ are equivalent iff there is a bijection $$f: S \to T$$ between the elements of $$S$$ and those of $$T$$.

This can be written $$S \sim T$$.

Some sources use $$S \simeq T$$.

If $$S$$ and $$T$$ are not equivalent we write $$S \not \sim T$$.

Other terms that are used that mean the same things as equivalent are:
 * Equipotent (equalness of power), from which we refer to equivalent sets as "having the same power";
 * Equipollent (equalness of strength);
 * Equinumerous (equalness of number);
 * Similar.

Also see

 * Cardinality