Set Equivalence/Examples

Examples of Set Equivalence

Arbitrary Example $1$

Let:

\(\ds S\)

\(=\)

\(\ds \set {2, 4, 6, 8}\)

\(\ds T\)

\(=\)

\(\ds \set {3, 5, 7, 9}\)

There exists a bijection between $S$ and $T$, that is: $f : x \mapsto x + 1$, for example.

They both have the same cardinality, that is $4$.

Arbitrary Example $2$

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, 2, 3}\)

\(\ds B\)

\(=\)

\(\ds \set {1, 2, 3, 4, 5}\)

There does not exists a bijection between $A$ and $B$.

That is because they do not have the same cardinality.