TOPICS

Introduction

Relations and Functions

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q,
i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q }.If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = φ.

e.g., A = {a, b, c}, B = {x, y}
A x B = {(a, x), (a, y), (b, x), (b, y), (c, x), (c, y)}

A pair of elements grouped together in a particular order.

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.

(ii) If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq.

(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B.

(iv) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

e.g., A = {1, 2, 3, 4} and B = {a, b, c, d}
R = {(1, a), (2, b), (3, a), (4, c)}

The image of an element x under a relation R is given by y, where (x, y) ∈ R.

The relation R is the set of all first elements of the ordered pairs in a relation R is called domain.

The relation R is the set of all second elements of the ordered pairs in a relation R is called range.

The whole set of second elements is called the co-domain of the relation R. Note that range ⊆ codomain.

e.g., A = {1, 2, 3, 4} and B = {a, b, c, d}
R = {(1, a), (2, b), (3, a), (4, c)}

Here, Domain = 1, 2, 3, 4.

Range = a, b, c.

Co-domain = All set of B.

A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B.

We write f: A → B, where f(x) = y.

Here, A is the domain and B is the Co-domain of f.

The range of the function is the set of images.

If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the image of a under f and a is called the preimage of b under f.

A real function has the set of real numbers or one of its subsets both as its domain and as its range.

There have 5 algebra of real functions which are given below :

(i) Addition of two real functions

(ii) Substraction of two real functions

(iii) Multiplication by a scalar

(iv) Multiplication of two real functions

(v) Quotient of two real functions

Let f : X → R and g : X → R be any two real functions, where X ⊂ R. Then, we define (f + g): X → R by (f + g) (x) = f (x) + g (x), for all x ∈ X.

Let f : X → R and g: X → R be any two real functions, where X ⊂ R. Then, we define (f – g) : X → R by (f – g) (x) = f(x) – g(x), for all x ∈ X.

Let f : X → R be a real valued function and α be a scalar. Here by scalar, we mean a real number. Then the product α f is a function from X to R defined by (α . f) (x) = α f (x), x ∈X.

The product of two real functions f:X → R and g:X → R is a function fg:X → R defined by (fg) (x) = f(x) g(x), for all x ∈ X.

This is also called pointwise multiplication.

Let f and g be two real functions defined from X → R where X ⊂ R. The quotient of f by g denoted by f/g is a function defined by, (f/g)(x) = f(x)/g(x) provided g(x) ≠ 0, x ∈ X

There have 7 types of function which are given below :

(i) Identity function

(ii) Constant function

(iii)Polynomial function

(iv) Rational function

(v) Modulus function

(vi) Signum function

(vii) Greatest integer function

Let R be the set of real numbers. Define the real valued function f : R → R by y = f(x) = x for each x ∈ R. Such a function is called the identity function. Here the domain and range of f are R. The graph is a straight line as shown in Figure. It passes through the origin.

Define the function f: R → R by y = f (x) = c, x ∈ R where c is a constant and each x ∈ R. Here domain of f is R and its range is {c}. The graph is a line parallel to x-axis. e.g., f(x) = 4

A function f : R → R is said to be polynomial function if for each x in R, y = f (x) = a₀ + a₁x + a₂x² + ...+ aᵢxⁿ, where i is a non-negative integer and a₀, a₁, a₂,...,aᵢ ∈ R. e.g., f(x) = x² + 2x + 1.

Rational functions are functions of the type f(x)/g(x), where f(x) and g(x) are polynomial functions of x defined in a domain, where g(x) ≠ 0. e.g., f(x) = 1/x.

The function f: R → R defined by f(x) = |x| for each x ∈ R is called modulus function. For each non-negative value of x, f(x) is equal to x. But for negative values of x, the value of f(x) is the negative of the value of x.

The function f:R → R defined by is called the signum function. The domain of the signum function is R and the range is the set {–1, 0, 1}.

The function f: R → R defined by f(x) = [x], x ∈ R assumes the value of the greatest integer, less than or equal to x. Such a function is called the greatest integer function. e.g., [x] = –1 for –1 ≤ x < 0.

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