Larger Set has Larger Set Difference

Theorem

Let $U, V$ be finite sets.

Let $\card V < \card U$.

Then:

$\card {V \setminus U} < \card {U \setminus V}$

Proof

We have:

\(\ds \card {U \setminus V}\)

\(=\)

\(\ds \card U - \card {U \cap V}\)

Cardinality of Set Difference

\(\ds \)

\(>\)

\(\ds \card V - \card {U \cap V}\)

As $\card V < \card U$

\(\ds \)

\(=\)

\(\ds \card {V \setminus U}\)

Cardinality of Set Difference

$\blacksquare$