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

$\blacksquare$

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: $\text {(i)}$