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A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective.
A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences. [4]
For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not. An injection: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function. A surjection: a function that is surjective ...
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X 2, with R•S interpreted as the usual composition of binary relations R and S, and with the ...
. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind.
Binary operators accept two relations as input and combine them into a single output relation. For example, taking all tuples found in either relation , removing tuples from the first relation found in the second relation , extending the tuples of the first relation with tuples in the second relation matching certain conditions, and so forth.
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, [13] but not antitransitive. [14]
For example, the relation + + = defines y as an implicit function of x, called the Bring radical, which has as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and n th roots .