Empty Intersection iff Subset of Complement

Theorem

$S \cap T = \varnothing \iff S \subseteq \complement \left({T}\right)$

where:

• $S \cap T$ denotes the intersection of $S$ and $T$
• $\varnothing$ denotes the empty set
• $\complement$ denotes set complement
• $\subseteq$ denotes subset.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \cap T\)	\(=\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle S \cap \complement \left({\complement \left({T}\right)}\right)\)	\(=\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Complement of Complement
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle S\)	\(\subseteq\)	\(\displaystyle \complement \left({T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Intersection of Complement with Subset is Empty

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $1.6$