Set Difference with Disjoint Set

Theorem

Let $S, T$ be sets.

Then:

$S \cap T = \O \iff S \setminus T = S$

where:

$S \cap T$ denotes set intersection

$\O$ denotes the empty set

$S \setminus T$ denotes set difference.

$\ds S \cap T$

$=$

$\ds \O$

$\ds \leadstoandfrom \ \ $

$\ds S$

$\subseteq$

$\ds \map \complement T$

$\ds \leadstoandfrom \ \ $

$\ds S \cap \map \complement T$

$=$

$\ds S$

$\ds \leadstoandfrom \ \ $

$\ds S \setminus T$

$=$

$\ds S$

$\blacksquare$

1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.11$