Relations and Functions

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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(x1) = f(x2) ⇒ x1 = x2 

x1, x2 ∈ 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.
→  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

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Relations and Functions

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Relation A relation R from a non-empty set A to a non empty set B is a subset of the Cartesianproduct A × B.  The set of all first elements of the ordered pairs in a relation R from aset A to a set B is called the domain of the relation R.  The set of all second elements ina relation R from a set A to a set B is called the range of the relation R.  The whole setB is called the codomain of the relation R.  Note that range is always a subset ofcodomain. A relation R on set is the subset of A×A. Therefore,  R ⊂ A × A. Types of Relations Extreme Relation 1. Empty  A relation R in a set A is called empty relation, if no element of A is related to anyelement of A, i.e., R = φ ⊂ A × A. 2. Universal  A relation R in a set A is called universal  3. Reflexive  A relation R in A is said to be reflexive if aRa for all a∈A 4. Symmetric  R is symmetric ifaRb ⇒ bRa, ∀ a, b ∈ A 5. Transitive it is said to be transitive if aRb and bRc ⇒ aRc∀ a, b, c ∈ A ...
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