Set Difference is Subset

Theorem

Set difference is a subset of the first set:

$S \setminus T \subseteq S$

Proof 1

\(\ds x \in S \setminus T\)

\(\leadsto\)

\(\ds x \in S \land x \notin T\)

Definition of Set Difference

\(\ds \)

\(\leadsto\)

\(\ds x \in S\)

Rule of Simplification

The result follows from the definition of subset.

$\blacksquare$

Proof 2

\(\ds S \setminus T\)

\(=\)

\(\ds S \cap \complement_S \left({T}\right)\)

Set Difference as Intersection with Relative Complement

\(\ds \)

\(\subseteq\)

\(\ds S\)

Intersection is Subset

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8$
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets