Tests for Finite Set Equality

Theorem

Let $S$ and $T$ be finite sets.

Let $S \subseteq T$.

The following statements are equivalent:

$(a): \quad \card S = \card T$

$(b): \quad \card {T \setminus S} = 0$

$(c): \quad T \setminus S = \O$

$(d): \quad T = S$

Proof

We have:

\(\text {(a)}: \quad\)

\(\ds \card S\)

\(=\)

\(\ds \card T\)

\(\text {(b)}: \quad\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \card {T \setminus S}\)

\(=\)

\(\ds \card T - \card S = 0\)

Cardinality of Set Difference with Subset

\(\text {(c)}: \quad\)

\(\ds \leadstoandfrom \ \ \)

\(\ds T \setminus S\)

\(=\)

\(\ds \O\)

Cardinality of Empty Set

\(\ds \leadstoandfrom \ \ \)

\(\ds T\)

\(\subseteq\)

\(\ds S\)

Set Difference with Superset is Empty Set

\(\text {(d)}: \quad\)

\(\ds \leadstoandfrom \ \ \)

\(\ds T\)

\(=\)

\(\ds S\)

Definition of Set Equality