Cardinality of Complement

Theorem

Let $T \subseteq S$ such that $\card S = n, \card T = m$.

Then:

$\card {\relcomp S T} = \card {S \setminus T} = n - m$

where:

$\relcomp S T$ denotes the complement of $T$ relative to $S$

$S \setminus T$ denotes the difference between $S$ and $T$.

Proof

The result is obvious for $S = T$ or $T = \O$.

Otherwise, $\set {T, S \setminus T}$ is a partition of $S$.

Let $\card {S \setminus T} = p$.

Then by the Fundamental Principle of Counting:

$m + p = n$

and the result follows.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis: Theorem $19.2$