Symmetric Difference with Self is Empty Set

Theorem

The symmetric difference of a set with itself is the empty set:

$S \symdif S = \O$


Proof

This follows directly from Symmetric Difference of Equal Sets:

$S \symdif T = \O \iff S = T$

substituting $S$ for $T$.

$\blacksquare$


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $7 \ \text{(ii)}$
  • 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: $\text {(v)}$
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