Set Difference Operation

A set difference operation is a set operation that produces the set members that are in one set but not in another sets.

• AKA: CSO, Set Complement Function.
• Context:
  • It can (typically) be a Binary Set Operation expressed as A \ B.
  • It can range from being a Relative Set Difference Operation to being an Absolute Set Difference Operation, if the comparison set is assumed to be the Universe Set.
• Example(s):
  • RCSO({a,b,c,d,e}, {a,c,d}) ⇒ {b,e}.
  • {a,b,c,d,e} \ {a,c,d} ⇒ {b,e}.
  • …
• Counter-Example(s):
  • a Set Symmetric Difference.
  • a Union Set Operation.
  • an Intersection Set Operation.
• See: σ-Field, Complement Set Law.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Complement_(set_theory)

In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another. The terms "absolute" and "relative" complement refer to more specific applications of the concept, with universal complements referring to elements unique to the universal set and the latter referring to the unique elements of one set in relation to another.

Absolute Complement: If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of [math]\displaystyle{ A }[/math], and is denoted by A^C or sometimes A′, also the same set often is denoted by C_U [math]\displaystyle{ A }[/math] (or C A if U is fixed), that is:

A^C = U ∖ A.

Relative complement: If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A. The relative complement of A in B is denoted B ∖ A (sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b − a, where b is taken from B and a from A). Formally:

B \ A = { x ∈ B | x ∉ A }.

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