CHAPTER 1 - RELATIONS AND FUNCTIONS
Types of Relations
A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A
A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A A.
Equivalence relation
A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
A relation R in a set A is called
(i) reflexive, if (a, a) ∈ R, for every a∈ A,
(ii) symmetric, if (a1, a2) ∈ R implies that (a2, a1)∈ R, for all a1, a2 ∈ A.
(iii) transitive, if (a1, a2) ∈ R and (a2, a3)∈ R implies that (a1, a3)∈ R, for all a1, a2, a3 ∈ A.
Types of Functions
A function f : X → Y is defined to be one-one (or injective),
if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, f (x1) = f (x2) implies x1 = x2. Otherwise, f is called many-one.
A function f : X → Y is said to be onto (or surjective),
if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f (x) = y.
Definition 7
A function f : X → Y is said to be one-one and onto (or bijective ), if f is both one-one and onto.
COMPOSITE FUNCTIONS
