Set Difference is Subset

Theorem

Set difference is a subset of the first set:

$S \setminus T \subseteq S$

$\ds x \in S \setminus T$

$\leadsto$

$\ds x \in S \land x \notin T$

Definition of Set Difference

$\ds $

$\leadsto$

$\ds x \in S$

The result follows from the definition of subset.

$\blacksquare$

$\ds S \setminus T$

$=$

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

$\ds $

$\subseteq$

$\ds S$

$\blacksquare$

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