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 \(R = \phi \subseteq X \times X\).

  • Universal relation is the relation \(R\) in \(X\) given by \(R = X \times X\).

  • Reflexive relation \(R\) in \(X\) is a relation with \((a, a) \in R\ \forall a \in X\).

  • Symmetric relation \(R\) in \(X\) is a relation satisfying \((a, b) \in R \Rightarrow (b, a) \in R\).

  • Transitive relation \(R\) in \(X\) is a relation satisfying \((a, b) \in R\) and \((b, c) \in R\) implies that \((a, c) \in R\).

  • Equivalence relation \(R\) in \(X\) is a relation which is reflexive, symmetric and transitive.

  • Equivalence class \([a]\) containing \(a \in X\) for an equivalence relation \(R\) in \(X\) is the subset of \(X\) containing all elements related to \(a\).

  • A function \(f : X \rightarrow Y\) is one-one (or injective) if
    \[ f(x_1) = f(x_2) \Rightarrow x_1 = x_2,\ \forall\ x_1, x_2 \in X. \]

  • A function \(f : X \rightarrow Y\) is onto (or surjective) if given any \(y \in Y\), \(\exists\ x \in X\) such that \(f(x) = y\).

  • A function \(f : X \rightarrow Y\) is one-one and onto (or bijective), if \(f\) is both one-one and onto.

  • Given a finite set \(X\), a function \(f : X \rightarrow X\) 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.

Last modified: Saturday, 9 August 2025, 7:30 PM