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In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complement of A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.
If A and B are sets, then the relative complement of A in B,^{[1]} also termed the set-theoretic difference of B and A,^{[2]} is the set of elements in B, but not in A.
The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard (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
Examples:
The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.
If A, B, and C are sets, then the following identities hold:
[ Alternately written: A ∖ (B ∖ C) = (A ∖ B)∪(A ∩ C) ]
If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of A, and is denoted by A^{c} or sometimes A′. The same set often^{[3]} is denoted by \complement_U A or \complement A if U is fixed, that is:
For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of even numbers.
The following lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.
If A and B are subsets of a universe U, then the following identities hold:
The first two complement laws above shows that if A is a non-empty, proper subset of U, then {A, A^{c}} is a partition of U.
In the LaTeX typesetting language, the command \setminus^{[4]} is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. A variant \smallsetminus is available in the amssymb package.
\setminus
\backslash
\mathbin{\backslash}
\smallsetminus
Some programming languages allow for manipulation of sets as data structures, using these operators or functions to construct the difference of sets a and b:
a
b
a.Except(b);
set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());
(clojure.set/difference a b)
set-difference, nset-difference
diff = a - b
difference a b
a \\ b
diff = a.clone();
setdiff
Complement
Set.S.diff
SetDifference := a - b;
#for perl version >= 5.10
@a = grep {not $_ ~~ @b} @a;
$A ∖ $B
$A (-) $B # texas version
array_diff($a, $b);
a(X),\+ b(X).
diff = a.difference(b)
diff = a—b
a difference: b
SELECT * FROM A
EXCEPT SELECT * FROM B
comm -23 a b
grep -vf b a
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