Symmetric Difference with Empty Set
Theorem
- $S \symdif \O = S$
where $\symdif$ denotes the symmetric difference.
Proof
| \(\ds S \symdif \O\) | \(=\) | \(\ds \paren {S \cup \O} \setminus \paren {S \cap \O}\) | Definition 2 of Symmetric Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds S \setminus \paren {S \cap \O}\) | Union with Empty Set | |||||||||||
| \(\ds \) | \(=\) | \(\ds S \setminus \O\) | Intersection with Empty Set | |||||||||||
| \(\ds \) | \(=\) | \(\ds S\) | Set Difference with Empty Set is Self |
$\blacksquare$
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $6$
- 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: $\text {(iii)}$