## Relations and Functions

### Sets and Relation Class 11

Let us begin with some basics of sets and relation class 11, before learning about the difference between relation and function in Mathematics. A set is defined as a group of objects or elements. We have learned different types of sets such as empty set, equal set, subset, or power set in our earlier classes. In this article, we will learn about Cartesian products of sets as it will help you to solve the questions based on sets and relations class 11.

### Cartesian Products of Sets

Let us assume there are two empty sets M and N. So the Cartesian product of M and N will be the set of each ordered pair of elements from M and N.

M x N = {(A, B}): a Є M, n Є N})

Let M = {m₁, m2, m3, m4} and N = {n₁, n₂}

Hence, the Cartesian product of M and N will be,

M x N = {m₁n₁, mn1, mn1, m4n, m1n2, m1n2, m1n3, m₁n4}

For Example: Let us take X = (a, b, c) and Y = (1, 2, 3)

Hence product of X and Y = (a₁, a₂, a₃, b₁, b₂, b₃, c₁, c₂. c₃)

The above set has 8 ordered pairs.

Two ordered pairs X and Y will only be equivalent if the corresponding first element and second element will be equivalent to each other.

### Relation and Function Class 11 Explain

Relation

A relation M is the subset of Cartesian Product of M and N, where M and N are considered as two nonempty sets. It is concluded by stating their relationship between the first and second element of the ordered pair. The set of all the first elements of the ordered pair is known as domain M whereas the set of all the second elements of the ordered pair is known as the range of M.

Example-1. (2, 1), (5, 7),(9,7), usually written in set notation with curly brackets.

Example-2. Let us take two non empty sets M = {a, b, c} and N {d, e}.Find the number of relations from M to N.

Solution = M x N = {(a, d), (a, e), (a,f), (b, d), (b, e), (b, f), (c, d), (c.e), (c, f)}

Number of subsets, x (M×N) = 26

Hence, the number of relations from M to N is 26

### Function Class 11

Here,we will brief functions class 11.

A function is a relation only if each element of non-empty set M, has only one range to a nonempty set N.

For Example- Let M and N are two non-empty sets, mapping from M to N will be considered as a function only when each element in set M has only one image in set N

### Solved Examples

If set M has 3 elements and the set N = (4, 5, 6), then find the number of elements in (M×N)?

Solution:

Total Number of elements in set M = 3 elements

Total number of elements in set N= 3 elements

Number of elements in set M ×N

= Number of elements in set M×Number of elements in set N

= 3 * 3

=9

Hence, the number of elements in (M x N) is 9

If M= {a, b, c} and N ={r}, find the set M x N, Are these products equal?

Solution: M= {a, b, c}

And N ={r}

M* N = {a, b, c} * {r}

M* N = {a, r}, {b, r}, {c, r}

N* M ={r} * {a, b, c}

N* M = {r, a}, {r, b}, {r, c}

As, {a, r}, ={r, a},

M× N = N× M

Since the corresponding first element of and the second element of two sets are not equal.

The above two ordered pairs M* N are not equal as the corresponding first element and second element of two sets are different.

### Facts

• The modern definition of the functions was first given by German Mathematician Peter Dirichlet in 1837.

• Functions are ubiquitous in mathematics and are essential for formulating physical relationships in Science.

### Quiz Time

1. The number of subsets of a set containing n elements is

a. n

b. 2n -1

c. n₂

d. 2n

2. Let A = { 1,2,3) and B = {6.7} ,Find A * B

a. {( 1,6), { 1,7) , ( 2,6), ( 2, 7), ( 3,7),(3,6)}

b. { 1,2,3,6,7}

c. {(1,6), (2,6), (3,6)}

d. {( 6,1), (6,2), ( 6,3), (7,1), (7,2), (7,3)}

1. Explain types of relations and Function in Mathematics.

Types of Function class 11

One to A function f: MN is One to One if for each element o M if there is a different element of N. It is also known as injective, For example- If a₁ Є M and a₂ Є N, f is defined as f: M – N such that f (a₁) =f (a₂)

Onto function or Surjective

A function for which each element of set M, there exist a pre-image in set N (image will be uploaded soon)

One–one and Onto Function or Bijective Function

The function f matches with every element of A with a discrete element of B and each element of B has a preimage in A (image will be uploaded soon)

Many to one Function

A function which maps two or more element of A to the same element of set B (image will be uploaded soon)

Types of Relation

Equivalence Relations

A relation will be known as equivalence only if a relation is reflexive, symmetric and transitive.

Transitive Relation

If (p, q) Є M, (q, r,) Є M then (p,r) Є M, for all p,q,r Є and this relation in set M is a transitive relation.

Symmetric Relation

A symmetric relation is a relation R on a set A if (p, q) Є R then (q,p) Є R, for all p and q Є M.

Reflexive Relation

A relation will be known as reflexive if every element of set M maps to itself I.e. for every p Є M, (p,p) Є N

When every element of set M will relate to itself only, it will be known as an identity relation.

M = (A, A}), Є a

Inverse Relation

If M is a relation from set M to set N i.e. Є M*N. The relation M ‾1 = {b,a): (a, b) Є R}.

Universal Relation

A relation M in a set, say P is a universal relation if every element of P is related to each element of P, i.e. M = P*P/It is also known as a full relation.

2. What is the Difference Between Relation and Function?

Differences between relation and function are explained below in tabular form.

 Relation Function A relation is a subset of the Cartesian product A function will be considered as a relation if there will be only one output for each input. A relation is symbolized as “ R” A function is symbolized as “F”  or “f” Every relation is not considered as a function Every function will be considered as a function For Example- (1, x), (3, y), (1,z)   It will not be considered as a function as “1” is an input for both x and z For Example- {(1, x), (3, y), (5,x)}