In this chapter, we studied different types of relations and equivalence relation,
composition of functions, invertible functions and binary operations. The main features
of this chapter are as follows:

  • Empty relation is the relation R in X given by R = φ ⊂ X × X.
  • Universal relation is the relation R in X given by R = X × X.
  • Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.
  • Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
  • Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.
  • Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
  • Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.
  • A function f : X → Y is one-one (or injective) if
  • $f(x_1) = f(x_2) ⇒ x_1= x_2 ∀ x_1, x_2 ∈ X$
  • A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
  • A function f : X → Y is one-one and onto (or bijective), if f 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)) ∀ x ∈ A.
  • A function f : X → Y is invertible if ∃ g : Y → X such that $gof = I_X$ and $fog = I_Y$
  • A function f : X → Y is invertible if and only if f is one-one and onto.
  • Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set
  • A binary operation ∗ on a set A is a function ∗ from A × A to A.
  • An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈X.
  • An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where, e is the identity for the binary operation ∗. The element b is called inverse of a and is denoted by $a^{–1}$
  • An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.
  • An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c)∀ a, b, c in X.


Historical Note
The concept of function has evolved over a long period of time starting from
R. Descartes (1596-1650), who used the word ‘function’ in his manuscript
“Geometrie” in 1637 to mean some positive integral power $x^n$ of a variable xwhile studying geometrical curves like hyperbola, parabola and ellipse. James Gregory (1636-1675) in his work “ Vera Circuli et Hyperbolae Quadratura” (1667) considered function as a quantity obtained from other quantities by successive use of algebraic operations or by any other operations. Later G. W. Leibnitz (1646-1716) in his manuscript “Methodus tangentium inversa, seu de functionibus” written in 1673 used the word ‘function’ to mean a quantity varying from point to point on a curve such as the coordinates of a point on the curve, the slope of the curve, the tangent and the normal to the curve at a point. However, in his manuscript “Historia” (1714), Leibnitz used the word ‘function’ to mean quantities that depend on a variable. He was the first to use the phrase ‘function of x’. John Bernoulli (1667-1748) used the notation φx for the first time in 1718 to indicate a function of x. But the general adoption of symbols like f, F, φ, ψ ... to represent functions was made by Leonhard Euler (1707-1783) in 1734 in the first part of his manuscript “Analysis Infinitorium”. Later on, Joeph Louis Lagrange (1736-1813) published his manuscripts “Theorie des functions analytiques” in 1793, where he discussed about analytic function and used the notion f (x), F(x), φ(x) etc. for different function of x. Subsequently, Lejeunne Dirichlet (1805-1859) gave the definition of function which was being used till the set theoretic definition of function presently used, was given after set theory was developed by Georg Cantor (1845-1918). The set theoretic definition of function known to us presently is simply an abstraction of the definition given by Dirichlet in a rigorous manner.


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