“Relations and Functions” are two vital topics in algebra. You cannot skip these topics in the formative classes. These two topics build a great foundation in algebra. The following example offers a great way to understand how relations and functions are defined in mathematics. 

 

Ordered pair of sets is denoted as (INPUT, OUTPUT): 

 

A relation exhibits the relationship between INPUT and OUTPUT. 

On the other hand, a function is a special relation that derives one OUTPUT for each given INPUT.

 

It should be duly noted that all functions are relations, but not all relations are functions.

 

To understand the Cartesian product of sets, first suppose that there are two non-empty sets C and D. The Cartesian product of C and D can be denoted as the set of all ordered pairs of elements from C and D.

 

C × D = {(c,d) : c ∊ C, d ∊ D}

 

Let C = {c1, c2, c3, c4} and D = {d1, d2}

 

The cartesian product of sets C and D is given by;

 

C × D = {(c1, d1), (c2, d1), (c3, d1), (c4, d1), (c1, d2), (c2, d2), (c3, d2), (c4, d2)}

 

Example: Let us say, C = {c, d, e} and D = { 4,5,6}

 

Therefore, C × D = {(c, 4), (c, 5), (c, 6), (d, 4), (d, 5), (d, 6), (e, 4), (e, 5), (e, 6)}.

 

This Cartesian product set (CxD) has nine ordered pairs. The product set can also be represented in a tabular form.

 

Two ordered pairs, C and D, are equal only when the corresponding first and second elements are equal.

 

Relation R' is the subset of the Cartesian product of C x D, where C and D are two non-empty elements, the correlation between the first and second elements of the ordered pair of C × D. The set of all primary elements of the ordered pairs is known as the domain of R, and the set of all second elements of the ordered pairs is known as the range of R'.

 

For two sets C = {a, b, c} and D = {axe, box, crow}, the Cartesian product have nine ordered pairs, given by;

 

C × D = {(a, Axe), (a, box), (a, crow), (b, Axe), (b, box), (b, crow), (c, Axe), (c, box), (c, crow)}

 

A subset of C x D can be obtained by introducing a relation R, between the elements of C and D as:

 

R' = {(a,b) : a is the first letter of word b, a ∊ C, b ∊ D}

 

The relation between C and D can be represented as;

R' = {(a,Axe),(b,box),(c,crow)}

 

Example: Let C={a,b} and D = {c, d}. Find the number of relations from C to D.

 

Solution: C × D = {(a,c),(a,d),(b,c),(b,d)}

 

Number of subsets, n (C × D) = 24 . 

So, the total number of relations from set C to set D is 24.

 

Different types of relations are as follows:

 

• • Empty Relations

  • Identity Relations

  • Universal Relations

  • Reflexive Relations

  • Inverse Relations

  • Transitive Relations

  • Symmetric Relations

 

A function states that there should be only one output for each input (or) a special kind of relation, which follows a rule, i.e. every x-value (input value) should be connected to only one y-value (output value).

If ‘F’ is the function from C to D and (x,y) ∊ f, then f(x) = y, where y is the image of x, under function f and x is the preimage of y, under ‘F'. It can be denoted as;

 

f: C → D.

 

Example: Let N be the set of natural numbers, and the relation R be defined as:

 

R = {(a,b) : b=a2, a,b ∈ N}. Is R a relation function or not?

 

Solution: The relation R = {(c,d) : d=c², a,b ∈ N}, we can see for every value of natural number, there is only one image. For example, if c=1 then d =1, if c=2 then d=4 and so on.

 

R is the relation function here.

 

Ques. If the set X has three elements and the set Y = {2, 4, 6}, then find the number of elements in (X × Y)?

Solution:

Given, set X has three elements, and the elements of set Y are {2, 4, and 6}.

So, the number of elements in set Y = 3

Then, the number of elements in (X × Y) = (Number of elements in X) × (Number of elements in Y)

= 3 × 3 = 9

Therefore, the number of elements in (X × Y) will be 9.

 

Ques. If G = {4, 8} and H = {5, 4, 2}, find G × H and H × G.

Solution:

Given, G = {4, 8} and H = {6, 4, 2}

The Cartesian product of two non-empty sets C and D are given as:

C × D = {(c, d): c ∈ C, d ∈ D}

We get, 

G × H = {(4, 6), (4, 4), (4, 2), (8, 6), (8, 4), (8, 2)}

H × G = {(6, 4), (6, 8), (4, 4), (4, 8), (2, 4), (2, 8)}

 

Ques. If X × Y = {(a, x), (a, z), (b, x), (b, z)}. Find X and Y.

Solution: 

X × Y = {(a, x), (a, z), (b, x), (b, z)}

The Cartesian product of two non-empty sets C and D is given as:

C × D = {(c, d): c ∈ C, d ∈ D}

Hence, X is the set of all first elements, and Y is the set of all second elements.

Therefore, X = {a, b} and Y = {x, z}

 

Ques. Let X and Y be two sets where n(X) = 3 and n (Y) = 2. If (w, 1), (y, 2), (a, 2) are in X × Y, find X and Y, where x, y, and a are distinct elements.

Solution:

Given,

n(X) = 3 and n(Y) = 2; and (w, 1), (y, 2), (a, 2) are in X × Y.

We know that,

X = Set of first elements of the ordered pair elements of X × Y.

Y = Set of second elements of the ordered pair elements of X × Y.

So, w, y, and z are the elements of X; and

1 and 2 are the elements of Y.

As n(X) = 3 and n(Y) = 2, it is clear that set X = {w, y, a} and set Y = {1, 2}.

 

Ques. Define a relation R on the set N of natural numbers by R = {(c, d): d = c + 2, c is a natural number less than 5; c, d ∈ N}. Depict this relationship using roster form. Write down the domain and the range.

Solution:

The relation R is given by:

R = {(c, d): d = c + 2, c is a natural number less than 5, c, d ∈ N}

The natural numbers less than 5 are 1, 2, 3, and 4.

So,

R = {(1, 3), (2, 4), (3, 5), (4, 6)}

We have,

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

Hence, Domain of R = {1, 2, 3, 4}

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

Hence, Range of R = {3, 4, 5, 6}

How many exercises are there in Chapter 2 Maths Class 11?

There are in total three exercises and one miscellaneous exercise in NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions

 

How can you find which relation is a function in Chapter 2 ofNCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions?

The unique relations (set of ordered pairs) which follow the rule where every x-value is associated with only one y-value is called a function.

 

Is NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions important?"

Yes, the topic Relations and Functions Class 11 is important for all Class 11 students. Many questions are asked from the chapter, making it vital for good marks in this exam and competitive exams.