Set Difference with Self is Empty Set

Theorem

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

$S \setminus S = \O$


Proof

From Set is Subset of Itself:

$S \subseteq S$

From Set Difference with Superset is Empty Set‎ we have:

$S \subseteq T \iff S \setminus T = \O$

Hence the result.

$\blacksquare$


Also see


Sources

  • 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 8 \ \text{(a)}$
  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.6$. Difference and complement: : Example $19$
  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \ \text{(d)}$
  • 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.17$
  • 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.5 \ \text{(ii)}$
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