User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 3

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Theoren

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

$\blacksquare$