Introduction of Cartesian products of sets

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)}

Ordered pair

A pair of elements grouped together in a particular order.

Note :

(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.

Relations

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)}

Image

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

Domain

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

Range

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

Co-domain

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

Example of Domain, Range and Co-domain

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.

Functions

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.

Image and pre-image

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.

Real function

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

Algebra of real functions

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

Addition 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.

Substraction 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.

Multiplication by a scalar

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.

Multiplication of two real functions

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.

Quotient of two real functions

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

Types of Function in graphical form

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

Identity 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.

Constant Function

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

Polynomial Function

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 Function

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.

Modulus Function

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.

Signum Function

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}.

Greatest Integer Function

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.