Relations and Functions

A relation R in a set A is called universal relation, if each element of A is related to every element of A. A relation R in a set A is called empty relation, if no element of A is related to any element of A. Both the empty relation and the universal relation are called trivial relations. A relation R in a set A is called an identity relation if each element in A is related to itself only. Let R be a relation on a non-empty set A then R is called a reflexive relation iff a R a for all ‘a’ belong to A. A relation R in a set A is said to be an equivalence relation iff R is a) reflexive, b) symmetric, and c) transitive. Let f be a function defined on the set A. If the images of distinct elements of A are distinct, then the function f is known as one-one function. If the images of the distinct elements of A are the same, then the function f is known as many-one function Let f be a function from the set A to set B. If for every y belongs to B, there exists an element x in A such that f(x) = y, then f is known as onto function. A function f from A to B is called into iff there exists at least one element in B, which is not the image of any element of A. A function f from A to B is called bijective or one - one onto if it is both one - one and onto. The composition of functions f: A → B and g : B → C is the function gof : A → C given by gof(x) = g(f(x)). A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = Ix and fog = Iy. Let S be a non-empty set. A function *: S✕S→S is called a binary operation on the set S.

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