1. Ordered Pair: A pair having one element as the first and the other as the second is called an ordered pair. An ordered pair having a as the first element and b as the second element. It is denoted by (a,b).

An ordered pair (a,b) is generally not the same as the ordered pair (b,a). But this will happen so when the two elements are identical. Thus, (5,6) different from (6,5); but (2,2) is the same as (2,2).

Two ordered pairs (a,b) and (c,d) are said to be equal if and only if a=c and b=d.

Example: If (2x-1, -3) = (3, y+3), then (2072 Questions)
a. x = 1, y = 0 b. x= – 1, y = – 3
c. x = 2, y = – 6 d. x = 0, y = – 1

solution: Given, (2x-1, -3) = (3, y+3)

from the definition of ordered pair

2x-1 = 3 & -3 = y+3

2x=3+1 -3-3=y

x=2 y=-6

(x,y)=(2,-6)

Hence, the correct answer is option c. x=2, y= -6

2. Cartesian Product: Given two sets A and B, the set of all ordered pairs(a,b) such that a∈A and b∈B is called the cartesian product of A and B. It is denoted by AxB. It is read as A cross B.

In the set-builder notation, we have

AxB={(a,b):a∈A,b∈B}

Example: The cartesian product of A={1,2} and B={2,3,4}

AxB={(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)}

It is different from the cartesian product of B={2,3,4} and A={1,2}

i.e. BxA={(2,1),(2,2),(3,1),(3,2),(4,1),(4,2)}

Note:

i). In general, AxB ∉ BxA

ii). If m is the number of elements in A and n is the number of elements in B, then the number of elements in AxB or BxA is mn

iii). If ℛ is the set of real numbers, then the cartesian product of ℛ on ℛ.

i.e ℛxℛ or R² is the set {(x,y):x∈ℛ, y∈ℛ}.

This cartesian product is represented by the entire cartesian coordinates plane.

Relation: Any subset of a cartesian product AxB of two sets A and B is called a relation. A relation from a set A to a set B is denoted by xℛy if x∈A and y∈B or simply by ℛ if (x,y) ∈ ℛ.

A relation from a set A to itself is called a relation on A. Relations may be expressed in various ways. Here are the some examples.

A={1,2} and B={2,3,4}

AxB={(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)}

ℛ={(2,2),(2,4)} ⊂ AxB, is a relation from A to B

Note:

If ℛ is a relation from A to B then

i). It is not necessary that every element of A has ℛ relation with some element of B.

ii). It is also not necessary that if aℛb and b is unique

iii). If B = A then ℛ⊂AxA and in such a case we read it as ‘ℛ is a relation on A’

iv). If ℛ = AxA, then ℛ is called universal relation on A.

v). ℛ= ɸ is called void relation on a set A.

Domain :The domain of a relation ℛ is the set of all first members of the pairs (x,y) of ℛ. It is denoted by Dom(ℛ).

Symbolically, Dom(ℛ) ={x:(x,y) ∈ ℛ for some y ∈ B}

Example: R={(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)} ⊂ AxB

Domain={1,2}

Range:The range of a relation ℛ is the set of all second members of the pairs (x,y) of ℛ. It is denoted by Ran(ℛ).

Symbolically, Ran(ℛ) ={x:(x,y) ∈ ℛ for some x ∈ A}

Example: R={(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)} ⊂ AxB

Range={2,3,4}

Inverse Relation: An inverse relation is the inverse of a given relation obtained by Interchanging or swapping the elements of each ordered pair. In other words, if (x, y) is a point in a relation ℛ, then (y, x) is an element in the inverse relation R^–1.

A relation ℛ from set A to B is a subset of the Cartesian product of A and B. ℛ is a subset of A × B. The elements of ℛ of the form of an ordered pair (a, b) where a ∈ A and b ∈ B.

The inverse relation of ℛ is denoted by R^–1. R^–1 is a subset of B × A. The elements of R^–1  of the form of an ordered pair (b, a) where b ∈ B and a∈ A.

Given a Relation: ℛ={(x,y): x ∈A,y∈B} ⊂ AxB

then its inverse is given by; R^–1 = {(y,x): y∈B,x ∈A}

Example: ℛ={(1,2),(1,3),(1,4),(2,2),(2,3),(2,4)}, then R^–1 ={(2,1),(3,1),(4,1),(2,2),(3,2),(4,2)}

Note: For Inverse Relation Simply interchanging the value of x & y

Types of Relation: The different types of Relation are describe below;

a. Reflexive Relation: A relation ℛ on a set A is reflexive if every element is related to itself.

Mathematically:

∀a∈A, (a,a)∈ℛ

Example: On the set A={1,2,3}, the relation ℛ={(1,1),(2,2),(3,3),(1,2),(2,1)} is reflexive.

b. Symmetric Relation: A relation ℛ on a set A is symmetric if whenever (a,b)∈ℛ, then (b,a)∈ℛ as well.

Mathematically: ∀a,b∈A, (a,b)∈ℛ⟹(b,a)∈ℛ

Example: For the set A={1,2,3}, the relation ℛ={(1,2),(2,1),(3,3)} is symmetric.

c. Anti-symmetric Relation: A relation ℛ on a set A is anti-symmetric if whenever (a,b)∈ℛ and (b,a)∈ℛ, then a=b.

Mathematically: ∀a,b∈A, (a,b)∈ℛ and (b,a)∈ℛ⟹a=b

Example: For A={1,2,3}, the relation ℛ={(1,2),(2,1),(2,2)} is anti-symmetric.

d. Transitive Relation: A relation ℛ on a set A is transitive if whenever (a,b)∈ℛ and (b,c)∈ℛ, then (a,c)∈ℛ.

Mathematically: ∀a,b,c∈A, (a,b)∈ℛ and (b,c)∈ℛ⟹(a,c)∈ℛ

Example: For A={1,2,3}, the relation ℛ={(1,2),(2,3),(1,3)} is transitive.

e. Equivalence Relation: A relation ℛ on a set A is an equivalence relation if it is reflexive, symmetric, and transitive.

Example: Equality == is an equivalence relation on any set because it is reflexive, symmetric, and transitive.

Function: Let A and B be two non-empty sets. A function f from A to B is a set of ordered pairs with the property that for each element x in A there is an unique element y in B. The Set A is called the domain of the function and the set B is called co-domain. If(x,y) ∈ f, it is customary to write y=f(x), y is called image of x and x is a pre-image of y.The set consisting all the images of the elements of a under the function f is called range of f. It is denoted by f(A)

1. Domain of the Function: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all values that can be plugged into the function without causing any undefined behavior.

If f is a function defined by f:A→B, where A is a set of possible inputs and B is a set of possible outputs, then, The domain of f is the set A, which consists of all elements for which f(x) is defined.

Shortcut Method:

i). The domain of √(a^2 – x^2) is [-a,a]

ii). The domain of 1/ √(a^2 – x^2) is (-a,a)

iii). The domain of √(x^2 – a^2) is (-∞,a] U [a,∞)

iv). The domain of 1/√(x^2 – a^2) is (-∞,a) U (a,∞)

v). The domain of √{(x – a) (b-x)} when a<b is [a,b]

vi). The domain of 1/ √{(x – a) (b-x)} when a<b is (a,b)

Example: Find the domain of f(x)= √(2-x) is

solution: Clearly,2-x≥ 0

= 2≥x

= x≤2

Hence, the domain of f(x)=√(2-x) is (-∞,2]

2. Co-domain of the function: The set B is known as the co-domain of the function.

3. Range of the Function: The set of values of y=f(x)∈B for every x∈A is known as rage of the function f.It is denoted by R(f).

R(f)={y:y∈B,y=f(x) for all x∈A}

How to find range: First put y=f(x) by suitable substitution, find x in term of y. Then find all such y for which x is defined in the domain set of these values of y is called the range of f(x).

Example: Find the domain and range of y=f(x)=x^2 – 6x + 6

solution: given,

y=f(x)=x^2 – 6x + 6

The given function is a polynomial of degree two in x,y is defined for all x ∈ℛ,so domain of

f=dom(f)=ℛ = (-∞,∞)

Again,

y=x^2 – 6x + 6

y+3=(x-3)^2

y= -3 +(x-3)^2

since, (x-3)^2≥ 0 so for all x ∈ℛ,y≥-3

Hence, range of y=f(x)=x^2 – 6x + 6 is [-3,∞)

4. Image: The element y∈B with which the element x∈A associates, is known as the image of x under f. It is also known as the value of f at x.

5. Pre-image: The element x∈A which associates with y∈B, is known as the pre-image of y under f.

6. Equal function: Two functions f and g are said to be equal i.e f=g if domain of f=domain of g and f(x)=g(x) for all x belonging to the domain of f(or domain of g).

Types of Functions: The types of function are give below

a. One-to one or Injective Function: A function f from a set A to another set B i.e. f: A → B is said to be one-one (1-1) or injective if distinct elements (or pre-images) in A have distinct images in B.
In symbols, for any x, y Є A,

In symbols, for any x, y Є A,

x ∉ y⇒ f(x) ∉ f(y);
f(x) = f(y) ⇒x= y.

or, equivalently,

In other words, a function f is said to be one-one or injective if (x, f(x)), (y, f(y)) Є ƒ⇒x= y.
Thus, under one-one function all elements of A are related to different elements of B.

b. Onto or Surjective Function: A function ƒ from a set A to another set B i.e., f: A → B is said to be onto or surjective, if every element of B is an image of at least one element of A, i.e., every element of B has a pre-image or, if ƒ (A) = B.
Sometimes such a function becomes a many-one onto function.

c. One to one correspondence or Bijective Function: A function that is both one-one and onto (i.e., injective and surjective) is called a bijective function. It is also known as a one-to-one correspondence.
In particular, a bijective function from a set A to itself is known as a permutation or operator on A.

d. Composition of Functions: If f: A → B and g: B → C be any two functions, then the composite function of f and g(also known as the product function or function of a function) is the function,

gof: A → C

defined by the equation (gof)(x)=g(f(x)).

The composition of two functions g and f is the new function we get by performing f first, and
then performing g. For example, if we let f be the function given by f(x) = x^2 and let g be the
function given by g(x) = x + 3, then the composition of g with f is called gf and is worked out
as
gf(x) = g(f(x)).

So we write down what f(x) is first, and then we apply g to the whole of f(x). In this case, if we
apply g to something we add 3 to it. So if we apply g to x^2, we add three to x^2. So we obtain

gf(x) = g(f(x)) = g(x^2) = x^2 + 3.

e. Odd Function: A function is an odd function if f(-x)= -f(x) for all x

f. Even Function: A function is an even function if f(-x)= f(x) for all x

g. Inverse/Invertible function: Let f: A → B be a one-one and onto function, then there exists a unique function g: B → A

Such that f(x)=y ⇔ g(y)=x, {g(y)=f^-1 (y) } ∀ xЄ A and yЄ B

Then, g is said to be inverse of f.

If any function f takes x to y then, the inverse of f will take y to x. If the function is denoted by f or F, then the inverse function is denoted by f^-1 or F^-1. One should not confuse (-1) with exponent or reciprocal here. 

Periodic Function: A function f(x) is said to be a periodic function of x, if there exist a positive real number T such that f(x+T)=f(x) for all x.

If period of f(x) is T then function 1/f(x), √f(x) are also function of same period.

If Period of f(x) is T then period of f(ax+b) is T/|a|

Example: Find the period function of sin(2x)

Given, sin(2x)= sin(2x+0)

we know, f(ax+b) then a=2 and b=0

Period of sin(2x) is T/|a|= 2π/2 = π

Hence, the Period of sin(2x) is π

From the Relations Functions and Graphs Chapter Video.