Domain and Range of a Relation

1

easyIf R = {(2, 3), (4, 5), (6, 7)}, what is the domain?

A. {2, 4, 6} B. {3, 5, 7} C. {2, 3, 4} D. {5, 6, 7}

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✔ Correct: A

The domain is the set of first elements of the ordered pairs: {2, 4, 6}.

2

easyIf R = {(1, 10), (2, 20), (3, 30)}, what is the range?

A. {1, 2, 3} B. {10, 20, 30} C. {1, 10, 20} D. {20, 30, 1}

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✔ Correct: B

The range is the set of second elements of the ordered pairs: {10, 20, 30}.

3

mediumLet A = {1, 2, 3, 4} and R = {(x, y) : y = x + 2}. Find the range.

A. {1, 2, 3} B. {2, 3, 4} C. {3, 4, 5, 6} D. {1, 2, 3, 4}

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✔ Correct: C

For x=1, y=3; x=2, y=4; x=3, y=5; x=4, y=6. Range = {3, 4, 5, 6}.

4

mediumIf R = {(x, x²) : x is a prime number less than 6}, find the range.

A. {4, 9, 25} B. {2, 3, 5} C. {1, 4, 9} D. {4, 9, 16}

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✔ Correct: A

Primes < 6 are {2, 3, 5}. Squares are {2², 3², 5²} = {4, 9, 25}.

5

hardLet R = {(x, y) : y = 1/x, x ∈ {1, 2, 4, 5}}. Find the range.

A. {1, 0.5, 0.25, 0.2} B. {1, 2, 4, 5} C. {0.1, 0.2, 0.4, 0.5} D. {1, 0.5, 0.2, 0.1}

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✔ Correct: A

y values are 1/1=1, 1/2=0.5, 1/4=0.25, 1/5=0.2.

6

hardIf R = {(x, y) : y = |x - 2|, x ∈ {0, 1, 2, 3, 4}}, find the range.

A. {0, 1, 2} B. {1, 2, 3} C. {0, 1, 2, 3} D. {0, 1, 2, 4}

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✔ Correct: A

|0-2|=2, |1-2|=1, |2-2|=0, |3-2|=1, |4-2|=2. Range = {0, 1, 2}.

7

very hardLet R = {(x, y) : x² + y² = 25, x, y ∈ {0, 3, 4, 5}}. Find the range.

A. {0, 3, 4} B. {3, 4, 5} C. {0, 3, 4, 5} D. {0, 4, 5}

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✔ Correct: C

Pairs are (0,5), (5,0), (3,4), (4,3). Range is {0, 3, 4, 5}.

1

Assertion (A): For R = {(1, 2), (2, 3)}, the domain is {1, 2}.
Reason (R): Domain is the set of all first elements of the ordered pairs.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both assertion and reason are true, and the reason correctly explains the assertion.

2

Assertion (A): For R = {(1, 2), (1, 3)}, the domain is {1, 1}.
Reason (R): A set cannot contain duplicate elements, so the domain is {1}.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false because sets do not contain duplicates; the domain is {1}.

1

mediumGiven A = {1, 2, 3, 4, 5, 6} and R = {(x, y) : y = x + 1}. Find: (i) R in roster form (ii) Domain (iii) Range.

Show Answer & Solution

(i) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
(ii) Domain = {1, 2, 3, 4, 5}
(iii) Range = {2, 3, 4, 5, 6}

2

mediumGiven R = {(x, y) : y = 2x, x ∈ {1, 2, 3, 4}}. Find: (i) The set of ordered pairs (ii) Domain (iii) Range.

Show Answer & Solution

(i) R = {(1, 2), (2, 4), (3, 6), (4, 8)}
(ii) Domain = {1, 2, 3, 4}
(iii) Range = {2, 4, 6, 8}

3

hardGiven R = {(x, y) : y = x², x ∈ {−3, −2, −1, 0, 1, 2, 3}}. Find: (i) R (ii) Domain (iii) Range.

Show Answer & Solution

(i) R = {(−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)}
(ii) Domain = {−3, −2, −1, 0, 1, 2, 3}
(iii) Range = {0, 1, 4, 9}

1

hardLet A = {1, 2, 3, 4, 5, 6}. Define a relation R on A by R = {(x, y) : y = x + 1}. (1) Write R in roster form. (2) Find the domain. (3) Find the range. (4) Find the codomain. (5) Is the range a subset of the codomain?

Show Answer & Solution

Step 1: Identify pairs where y = x + 1: (1,2), (2,3), (3,4), (4,5), (5,6).
Step 2: R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}.
Step 3: Domain = {1, 2, 3, 4, 5}.
Step 4: Range = {2, 3, 4, 5, 6}.
Step 5: Codomain = A = {1, 2, 3, 4, 5, 6}. Yes, {2, 3, 4, 5, 6} ⊆ {1, 2, 3, 4, 5, 6}.

2

very hardLet R = {(x, y) : x + 2y = 8, x, y ∈ N}. (1) Find the relation R. (2) Find the domain. (3) Find the range. (4) Verify if (2, 3) ∈ R. (5) Verify if (6, 1) ∈ R.

Show Answer & Solution

Step 1: Solve for y: y = (8 - x)/2. Since y ∈ N, x must be even.
Step 2: If x=2, y=3; x=4, y=2; x=6, y=1.
Step 3: R = {(2, 3), (4, 2), (6, 1)}.
Step 4: Domain = {2, 4, 6}, Range = {1, 2, 3}.
Step 5: (2, 3) ∈ R is true. (6, 1) ∈ R is true.

★

Case Study: A small shop tracks the sales of 5 items. The item IDs are {101, 102, 103, 104, 105}. The relation R between item ID (x) and price in hundreds (y) is given by y = x - 100.

Sub-question 1: (i) List the relation R as a set of ordered pairs. (ii) Find the domain.

Show Solution

(i) R = {(101, 1), (102, 2), (103, 3), (104, 4), (105, 5)}
(ii) Domain = {101, 102, 103, 104, 105}

Sub-question 2: (i) Find the range. (ii) If a new item 106 is added with price 6, does it belong to the same relation?

Show Solution

(i) Range = {1, 2, 3, 4, 5}
(ii) Yes, because 6 = 106 - 100, so (106, 6) satisfies the rule.

Definition of Domain and Range

Relation defined by an Equation

Properties of Relations

Section A: Multiple Choice Questions (MCQs)

Section B: Assertion–Reason Questions

Section C: Short Answer Questions

Section D: Long Answer Questions

Section E: Case-Based Question

CBSE Board & Sample Paper Past Year Questions

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