Symmetric Difference with Complement
Theorem
The symmetric difference of a set with its complement is the universal set:
- $S \symdif \relcomp {} S = \mathbb U$
Proof
| \(\ds S \symdif \relcomp {} S\) | \(=\) | \(\ds \paren {S \cup \relcomp {} S} \setminus \paren {S \cap \relcomp {} S}\) | Definition 2 of Symmetric Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {S \cup \relcomp {} S} \setminus \O\) | Intersection with Complement | |||||||||||
| \(\ds \) | \(=\) | \(\ds \mathbb U \setminus \O\) | Union with Complement | |||||||||||
| \(\ds \) | \(=\) | \(\ds \mathbb U\) | Set Difference with Empty Set is Self |
$\blacksquare$
Also see
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: $\text {(i)}$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(vii)}$