Relations & Functions Important Question

Very Short Answers [1 Marks]

Ques. State True or False.

“Every function is a relation but every relation is not a function”.

Ans. The above statement is true as every function is a relation but every relation is not a function.

Ques. How to identify Relations & Functions?

Ans. We can identify relations & functions as a binary operation is used to define the link between a set of elements in a relation. If each element of the first set is mapped to one and only one element of the second set, then it is considered as a function.

Ques. What are ordered pairs in Relations & Functions?

Ans. Relations & Functions have ordered pairs of the form (Input, Output). The input defines the domain and output defines the range of the relation/function.

Ques. Two sets are given: A={1, 2, 3, 5} and B={4, 6, 9}. Determine a relation R from set A to set B by R = {(x, y): the difference between x and y is odd; x∈A, y∈B}. Write relation R in roster form.

Ans. A = {1, 2, 3, 5}, B = {4, 6, 9} and R = {(x, y): the difference between x and y is odd; x∈A, y∈B}

Therefore, the relation R in roster form is {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}. 

Ques. Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.

Ans. It is given that R = {(x, x3): x is a prime number less than 10}

Prime numbers less than 10 from the above given condition are 2, 3, 5, 7

Therefore, R = {(2, 8), (3, 27), (5, 125), (7, 343)}

Ques. How can we represent Relations & Functions?

Ans. Relations & Functions can be represented in different forms like set-builder form, roster form, tabular form, arrow diagram, algebraic form and graphically.

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Short Answers [2 Marks Question]

Ques. Define Domain and Range.

Ans. Domain: Domain is defined as the set of all first elements of ordered pairs in a relation R from set A to set B. It is also called a set of inputs or pre-images.

Range - Range is defined as the set of all second elements of ordered pairs in a relation R from set A to set B. It is also called a set of outputs or images.

Ques. If relation G = {7, 8} and relation H = {5, 4, 2}, find the relation G×H and relation H×G.

Ans. It is given that G = {7, 8} and H = {5, 4, 2},

We know that cartesian product P×Q of two non-empty sets P & Q is defined as

P×Q = {(p, q): p∈P, q∈Q}

Therefore, 

G×H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

And H×G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

Ques. If A = {-1, 1}, find A×A×A.

Ans. For a non-empty set A, A×A×A is defined as

A×A×A = {(a, b, c): a, b, c ∈ A}

It is given in the question that, A = {-1, 1}

Therefore, 

A×A×A = {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)}

Ques. If relation A×B = {(a, x), (a, y), (b, x), (b, y)}, find the values of A and B.

Ans. It is given that A×B = {(a, x), (a, y), (b, x), (b, y)}

We know that cartesian product of two non-empty sets P & Q is defined as P×Q = {(p, q): p ∈ P, q ∈ Q}

Hence, it is clearly seen that A is the set of all first elements and B is the set of all second elements.

So, A = {a, b} and B = {x, y}

Ques. Let A = {1, 2, 3, 4, 6}. Let S be a relation on set A defined by 

{(a, b): a, b ∈ A, b is exactly divisible by a}

(i) Write S in roster form.

(ii) Find the domain of S

(iii) Find the range of S

Ans. It is given that A = {1, 2, 3, 4, 6} and S = {(a, b): a, b ∈ A, b is exactly divisible by a}

(i) S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)

(ii) Domain of S = {1, 2, 3, 4, 6}

(iii) Range of S = {1, 2, 3, 4, 6}

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Long Answers [3 Marks]

Ques. Find the domain and range of the real function f defined by f(x) = |x-1|.

Ans. The given real function is f(x) = |x-1|

It is clearly seen that |x-1| is defined for all real numbers

Therefore, Domain of f = R

Also, for x ∈ R, |x-1| assumes all real numbers

Hence, the range of f is the set of all non-negative real numbers.

Ques. Derive a relation R from A to A = {1, 2, 3, 4, 5, 6} as R = {(x, y): y = x + 1}. Find domain, codomain and range of R. 

Ans. We see that A = {1, 2, 3, 4, 5, 6} is the domain and codomain of R.

To find range, we need to calculate the values of y for each value of x, i.e. when x = 1, 2, 3, 4, 5, 6

x=1, y=1+1=2

x=2, y=2+1=3

x=3, y=3+1=4

x=4, y=4+1=5

x=5, y=5+1=6

x=6, y=6+1=7

As 7 doesn’t belong to A and relation R is defined on A, so x=6 has no image in A.

Therefore, Range of relation R = {2, 3, 4, 5, 6}

Domain = Codomain of relation R = {1, 2, 3, 4, 5, 6}

Ques. If (x/3+1, y-2/3) = (5/3, 1/3), then find the values of x and y.

Ans. It is given in the question that (x/3+1, y-2/3) = (5/3, 1/3)

As the ordered pairs are equal, the corresponding elements will also be equal.

So, x/3+1 = 5/3 and y-2/3 = 1/3

x/3+1 = 5/3

x/3 = 5/3-1 

x/3 = 2/3

x = 2

y-2/3 = 1/3

y = 1/2 + 2/3

y = 1

Therefore, x=2 and y=1.

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Very Long Answers [5 Marks]

Ques. The cartesian product A×A has 9 elements which are found (-1, 0) and (0, 1). Find the set A and remaining elements of relation A×A.

Ans. We know that, 

If n(A) = p and n(B) = q, then (A×B) = pq

So, n(A×A) = n(A)×n(A)

It is given that, n(A×A) = 9

n(A)×n(A) = 9

Hence, n(A) = 3

The ordered pairs given are (-1, 0) and (0, 1) which are two of the nine elements of A×A.

We know that, A×A = {(a, a): a∈A}

So, -1, 0 and 1 will be the elements of A.

Since, n(A) = 3, it is clear that A = {-1, 0, 1}

The remaining elements of set A×A are (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0) and (1, 1).

Ques. The function ‘t’ which maps temperature in degree celsius into temperature in degree Fahrenheit is defined as t(C) = 9C/5 + 32. Find (i) t(0)

(ii) t(28)

(iii) t(-10)

(iv) value of C when t(C) = 212

Ans. The given function is t(C) = 9C/5 + 32

So, 

(i) t(0) = 9×0/5 + 32 

= 0 + 32 = 32

(ii) t(28) = 9×28/5 + 32 = 252 + 160/5 

= 412/5 = 82.4

(iii) t(-10) = 9×(-10)/5 + 32 = 9×(-2) + 32

= -18 + 32 = 14

(iv) It is given that t(C) = 212

9C/5 + 32 = 212

9C/5 = 212-32 

9C/5 = 180

9C = 180×5

9C = 900

C = 900/9 

C = 100

Hence, the value of ‘t’ when the value of t(C) = 212 is 100. 

Ques. Which of the following relations are functions? Give reasons. If it is a function, then find its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Ans. (i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

As 2, 5, 8, 11, 14, 17 are elements of the domain of the given relation which have their unique images, so this relation is a function.

Domain = {2, 5, 8, 11, 14, 17} and 

Range = {1}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

As 2, 4, 6, 8, 10, 12, 14 are the elements of the domain of the given relation which have their unique images, so this relation is a function.

Domain = {2, 4, 6, 8, 10, 12, 14} and 

Range = {1, 2, 3, 4, 5, 6, 7} 

(iii) {(1, 3), (1, 5), (2, 5)}

As the first element i.e. 1 corresponds to two different images which are 3 and 5, so this relation is not a function.

As this relation is not a function, we will not be able to find the domain and range of the above relation.

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