Intersection with Relative Complement is Empty

Theorem

The intersection of a set and its relative complement is the empty set:

$T \cap \relcomp S T = \O$

Proof

\(\ds T \cap \relcomp S T\)

\(=\)

\(\ds \paren {S \setminus T} \cap T\)

Definition of Relative Complement

\(\ds \)

\(=\)

\(\ds \O\)

Set Difference Intersection with Second Set is Empty Set

$\blacksquare$

Also see

• Union with Relative Complement

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
• 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Exercise $1 \ \text {(b)}$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Theorem $3.2$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(h)}$