TOPICS

Introduction

Relations and Functions

A relation is a set of ordered pairs or, A relation between two sets is a collection of ordered pairs containing one object from each set.
**e.g.,** R = {(1, 2), (3, 4), (1, a), (5, b)}, R = {(x, y): y = x²} etc.

There have mainly four types of Relations : (i) Empty Relation, (ii) Universal Relation, (iii) Trivial Relation, (iv) Equivalence Relation

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
where, φ is empty set and A is any set.** e.g.,** R= φ ⊂ {2,3,4,5} x {2,3,4,5}.

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.
**e.g.,** R = {2,3,4,5} x {2,3,4,5}.

For some times both the empty relation and the universal relation are called trivial relations.

This Relation consits of three Relations. i.e., (i) Reflexive Relation (ii) Symmetric Relation (iii) Transitive Relation
and when a Relation is Reflexive, Symmetric and Transitive i.e., called Equivalence Relation. **e.g.,** R = {(x, y) : x and y work at the same place}.

A relation R in a set A is called Reflexive, if (a, a) ∈ R, for every a ∈ A.** e.g.,** R = {(1,1),(2,2),(3,3)}.

A relation R in a set A is called Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for all a, b ∈ A.** e.g.,** R = {(1,2),(2,1)}.

A relation R in a set A is called Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈ A.** e.g.,** R = {(1,2),(2,3),(1,3)}.

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let X and Y be two non empty sets. Let f be a rule so that corresponding to each x ∈ X, ∃ Unique y ∈ Y such that f(x) = y. Then this rule f is called function from X to Y and is written as f:X → Y. In this The set X is called the domain of f. The set of images is called range here range ={1,3} If y is image of x ∈X under function f then x is called pre image of y.

There have six types of Functions : (i) Injective Function or One-To-One Function, (ii) Surjective Function or On-To Function, (iii) Bijective Function (iv) Composition or Composite Function (v) Invertible Function.

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.

**e.g.,** If X={1,2,3}, Y={1,2,4,7,10}and f: X → Y defined by f(x)= x₂ then f(1)=1,f(2)=4,f(3)=4.

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.

**e.g.,** Let X={-1,0,1}, Y={A,B} and f: X → Y defined f(x)=|x| then function f is onto function.

A mapping is said to be a bijection or one- one and onto if it is both one-one and onto.

**e.g.,** Let X={1,2,3}, Y={2,4,6} and f:X → Y defined by f(x) =2x then f(1)=2, f(2)=4, f(3)=6.

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A → C given by gof(x) = g(f(x)), ∀ x ∈ A.

**e.g.,** If f: R → R and g: R → R be two functions defined respectively by
f(x) = x²+3x+1 & g(x) =2x-3 then gof is g(f(x)) = 2(x²+3x+1)-3.

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. The function g is called the inverse of f and is denoted by f^{-1}

**e.g.,** Let f : N → Y be a function defined as f(x) = 2x + 3
f(x) = y then, y = 2x + 3, x = (y-3)/2 i.e., f^{-1} = (y-3)/2.

A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b. **e.g.,** ∗ : R × R → R given by (a, b) → a + 4b² is a binary operation.

There have four types of Binary Opertaions : (i) Commutative Binary Operations, (ii) Associative Binary Operations, (iii) Identity Binary Operations, (iv) Invertible Binary Operations.

A binary operation ∗ on the set X is called commutative, if a ∗ b = b ∗ a, for every a, b ∈ X.

**e.g.,** a + b = b + a and a × b = b × a, ∀ a, b ∈ R, ‘+’ and ‘×’ are commutative binary operations.

A binary operation ∗ : A × A → A is said to be associative if (a ∗ b) ∗ c = a ∗ (b ∗ c), ∀a, b, c, ∈ A.

**e.g.,** (a + b) + c =a + (b + c) and (a x b) x c = a x (b x c) are associative binary operations.

Given a binary operation ∗ : A × A → A, an element e ∈ A, if it exists, is called identity for the operation ∗, if a ∗ e = a = e ∗ a, ∀ a ∈ A.

Given a binary operation ∗ : A × A → A with the identity element e in A, an element a ∈ A is said to be invertible with respect to the operation ∗,
if there exists an element b in A such that a ∗ b = e = b ∗ a and b is called the inverse of a and is denoted by a^{-1}.

CLASSES