$L\; :\; X\; to\; Y$ is a binary relation with $operatorname\{graph\},Lsubset\; Xtimes\; Y$

then the inverse relation is

$L^\{-1\}\; :\; Y\; to\; X$ defined by $y,L^\{-1\},xiff\; x,L,y$ ,

i.e. with $operatorname\{graph\},L^\{-1\}\; =\; \{(y,\; x)in\; Ytimes\; Xmid\; (x,\; y)\; in\; operatorname\{graph\},\; L\}$.

The inverse relation is also called the converse relation and may be written as $L^C$, $L^T$ (in view of its similarity with the transpose of a matrix), or $breve\{L\}$.

Properties

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, , a partial order, total order, strict weak order, (weak order), or an equivalence relation, its inverse is too.

However, if a relation is , this need not be the case for the inverse.

Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, e.g. , etc. (Parentheses would not be needed here but have been added for clarity.)

Inverse relation of a function

See also

• Bijection
• Function (mathematics)
• Inverse function
• Inverse relationship
• Relation (mathematics)