Union with Relative Complement

Theorem
The union of a set $T$ and its relative complement in $S$ is the set $S$:


 * $\complement_S \left({T}\right) \cup T = S$

Step 1
By the definition of relative complement, we have that:
 * $\complement_S \left({T}\right)\subseteq S$

and:
 * $T \subseteq S$

Hence by Union is Smallest Superset:
 * $\complement_S \left({T}\right) \cup T\subseteq S$

Step 2
Let $x \in S$.

By Law of Excluded Middle, one of the following two applies:


 * $(1): \quad x \in T$
 * $(2): \quad x \notin T$

If $(2)$, then by definition of relative complement:
 * $x \in S \setminus T = \complement_S \left({T}\right)$

So:
 * $x \in T \lor x \in \complement_S \left({T}\right)$

By definition of set union:
 * $x \in \complement_S \left({T}\right) \cup T$

Thus:
 * $x \in S \implies x \in \complement_S \left({T}\right) \cup T$

By definition of subset it follows that:
 * $S \subseteq \complement_S \left({T}\right) \cup T$

Step 3
From:
 * $\complement_S \left({T}\right) \cup T\subseteq S$

and:
 * $S\subseteq \complement_S \left({T}\right) \cup T$

it follows from definition of set equality that:
 * $S = \complement_S \left({T}\right) \cup T$