Symmetric Difference with Universe

Theorem

$\mathbb U \symdif S = \map \complement S$

where:

$\mathbb U$ denotes the universe

$\symdif$ denotes symmetric difference.

Proof

\(\ds \mathbb U \symdif S\)

\(=\)

\(\ds \mathbb U \cup S \setminus \mathbb U \cap S\)

Definition 2 of Symmetric Difference

\(\ds \)

\(=\)

\(\ds \mathbb U \cup S \setminus S\)

Intersection with Universe

\(\ds \)

\(=\)

\(\ds \mathbb U \setminus S\)

Union with Universe

\(\ds \)

\(=\)

\(\ds \map \complement S\)

Definition of Set Complement

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $9 \ \text{(i)}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): symmetric difference: $\text {(i)}$