Complement (set theory)

Relative complement

Definition

If A and B are sets, then the relative complement of A in B, also termed the set 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 ${\displaystyle B\setminus A}$ according to the ISO 31-11 standard. It is sometimes written ${\displaystyle B-A,}$ but this notation can be ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements ${\displaystyle b-a,}$ where b is taken from B and a from A.

Formally: $${\displaystyle B\setminus A=\{x\in B:x\notin A\}.}$$

Examples

• ${\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}.}$
• ${\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}.}$
• If ${\displaystyle \mathbb{R}}$ is the set of real numbers and ${\displaystyle \mathbb{Q}}$ is the set of rational numbers, then ${\displaystyle \mathbb{R}\setminus \mathbb{Q}}$ is the set of irrational numbers.

Properties

Let A, B, and C be three sets in a universe U. The following identities capture notable properties of relative complements:

• ${\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B).}$
• ${\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B).}$
• ${\displaystyle C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B),}$

  with the important special case ${\displaystyle C\setminus (C\setminus A)=(C\cap A)}$ demonstrating that intersection can be expressed using only the relative complement operation.

• ${\displaystyle (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).}$
• ${\displaystyle (B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).}$
• ${\displaystyle A\setminus A=\mathrm{\varnothing }.}$
• ${\displaystyle \mathrm{\varnothing }\setminus A=\mathrm{\varnothing }.}$
• ${\displaystyle A\setminus \mathrm{\varnothing }=A.}$
• ${\displaystyle A\setminus U=\mathrm{\varnothing }.}$
• If ${\displaystyle A\subset B}$, then ${\displaystyle C\setminus A\supset C\setminus B}$.
• ${\displaystyle A\supseteq B\setminus C}$ is equivalent to ${\displaystyle C\supseteq B\setminus A}$.

Complementary relation

A binary relation ${\displaystyle R}$ is defined as a subset of a product of sets ${\displaystyle X\times Y.}$ The complementary relation ${\displaystyle \overline{R}}$ is the set complement of ${\displaystyle R}$ in ${\displaystyle X\times Y.}$ The complement of relation ${\displaystyle R}$ can be written $${\displaystyle \overline{R}=(X\times Y)\setminus R.}$$ Here, ${\displaystyle R}$ is often viewed as a logical matrix with rows representing the elements of ${\displaystyle X,}$ and columns elements of ${\displaystyle Y.}$ The truth of ${\displaystyle aRb}$ corresponds to 1 in row ${\displaystyle a,}$ column ${\displaystyle b.}$ Producing the complementary relation to ${\displaystyle R}$ then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

LaTeX notation

In the LaTeX typesetting language, the command \setminus 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, but this symbol is not included separately in Unicode. The symbol ${\displaystyle \complement }$ (as opposed to ${\displaystyle C}$) is produced by \complement. (It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT.)