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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, m₂n1, m₃n1, m4n₁, 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

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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

                

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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)}