Sets, Relations and Functions — Exercise Solutions
1. How to Use This Page
This page covers every exercise question from TN Class 11 Maths Chapter 1 — Sets, Relations and Functions — with step-by-step worked solutions.
Study strategy:
| Step | Action | Why it works |
|---|---|---|
| 1 | Read Key Concepts to refresh definitions | Activates prior knowledge |
| 2 | Try each question on your own first | Builds problem-solving muscle |
| 3 | Check your answer against the worked solution | Immediate feedback loop |
| 4 | Note errors in Common Mistakes section | Prevents repeat errors |
| 5 | Use Exam Checklist before tests | Confidence boost |
Every numerical solution is worked out fully — no skipped steps.
2. Key Concepts (Quick Recap)
2.1 Set Notation
| Concept | Notation | Meaning |
|---|---|---|
| Belongs to | a ∈ A | a is an element of A |
| Does not belong to | a ∉ A | a is not an element of A |
| Subset | A ⊂ B | Every element of A is in B |
| Proper subset | A ⊊ B | A ⊂ B and A ≠ B |
| Superset | A ⊃ B | B is a subset of A |
| Empty set | ∅ or {} | Set with no elements |
| Universal set | U | Contains all elements under discussion |
| Power set | P(A) | Set of all subsets of A |
2.2 Set Operations and Formulas
| Operation | Notation | Formula / Definition |
|---|---|---|
| Union | A ∪ B | {x : x ∈ A or x ∈ B} |
| Intersection | A ∩ B | {x : x ∈ A and x ∈ B} |
| Difference | A − B | {x : x ∈ A and x ∉ B} |
| Complement | A′ or Aᶜ | U − A |
| Symmetric difference | A Δ B | (A − B) ∪ (B − A) |
2.3 Key Counting Formulas
2.4 De Morgan's Laws
2.5 Cartesian Product
For sets A and B:
2.6 Types of Relations
| Type | Condition |
|---|---|
| Reflexive | (a, a) ∈ R for all a ∈ A |
| Symmetric | (a, b) ∈ R ⟹ (b, a) ∈ R |
| Transitive | (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R |
| Equivalence | Reflexive + Symmetric + Transitive |
2.7 Types of Functions
| Type | Condition |
|---|---|
| One-one (Injective) | f(a) = f(b) ⟹ a = b |
| Onto (Surjective) | For every b ∈ B, ∃ a ∈ A such that f(a) = b |
| Bijection | Both one-one and onto |
3. I. Choose the Best Answer (MCQs)
Q1. If A = {1, 2, 3}, then the number of subsets of A is
(a) 4 (b) 6 (c) 8 (d) 9
The 8 subsets are: ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
Q2. If A = {1, 2} and B = {3, 4}, then A × B has how many elements?
(a) 2 (b) 4 (c) 8 (d) 16
Q3. If n(A) = 10, n(B) = 15, n(A ∪ B) = 20, then n(A ∩ B) is
(a) 5 (b) 10 (c) 15 (d) 25
Q4. Which of the following relations on A = {1, 2, 3} is transitive?
(a) R = {(1,1), (2,2)} (b) R = {(1,2), (2,1)} (c) R = {(1,2), (2,3), (1,3)} (d) R = {(1,2), (2,3)}
- (1,2) and (2,3) are in R, and (1,3) is also in R. The condition is satisfied.
- Option (d) fails because (1,2) and (2,3) are present but (1,3) is missing.
Q5. If f : A → B is one-one and onto, then f is called
(a) Injective (b) Surjective (c) Bijective (d) Constant
4. II. Fill in the Blanks
| 1. If A = ∅, then n(P(A)) = 1 (since P(∅) = {∅}, and | P(∅) | = 2⁰ = 1) |
|---|
2. A ∪ A′ = U (the universal set)
3. A ∩ A′ = ∅ (the empty set)
4. If n(A) = 5, then the number of proper subsets of A is 31 (2⁵ − 1 = 32 − 1 = 31, excluding A itself)
5. A × (B ∪ C) = (A × B) ∪ (A × C)
6. If f(x) = 2x + 3 and g(x) = x − 1, then (f ∘ g)(x) = 2x + 1
7. The identity relation on A = {a, b, c} is {(a,a), (b,b), (c,c)}
8. If A has 4 elements, then n(A × A) = 16 (4 × 4 = 16)
9. (A′)′ = A (complement of complement is the original set)
10. A − B = A ∩ B′ (set difference expressed using complement)
11. A ∪ ∅ = A (identity for union)
12. The number of relations from set A (with m elements) to set B (with n elements) is 2^(mn) (since A × B has mn ordered pairs, each may or may not be in the relation)
5. III. True or False (with corrections for false statements)
1. True or False: Every set is a subset of itself.
True. By definition, A ⊂ A for any set A (every element of A is in A).
2. True or False: A × B = B × A for all sets A and B.
False. Correction: A × B = B × A only if A = B or one of the sets is empty. In general, A × B ≠ B × A because ordered pairs have direction: (1, 3) ≠ (3, 1).
3. True or False: Every function is a relation.
True. A function f : A → B is a special type of relation from A to B where each a ∈ A has exactly one image in B.
4. True or False: An equivalence relation is always reflexive, symmetric, and transitive.
True. This is the definition of an equivalence relation.
5. True or False: If f is one-one, then f⁻¹ always exists as a function.
False. Correction: f⁻¹ exists as a function only if f is both one-one and onto (bijective). A one-one function that is not onto does not have an inverse function from B to A.
6. True or False: The empty set is a subset of every set.
True. ∅ ⊂ A for any set A because there is no element in ∅ that is not in A (vacuously true).
7. True or False: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
True. This is the distributive law of union over intersection.
6. IV. Match the Following
| Column A | Column B |
|---|---|
| 1. n(P(A)) where n(A) = 4 | (a) A ∩ B′ |
| 2. A − B | (b) 16 |
| 3. A Δ B | (c) (A − B) ∪ (B − A) |
| 4. f : R → R, f(x) = x² | (d) Neither one-one nor onto |
| 5. f : R → R, f(x) = 3x + 1 | (e) Bijective |
| 6. Reflexive relation | (f) (a, a) ∈ R for all a ∈ A |
Explanations:
- 1 → b: n(P(A)) = 2⁴ = 16
- 2 → a: A − B = {x : x ∈ A and x ∉ B} = A ∩ B′
- 3 → c: By definition, symmetric difference A Δ B = (A − B) ∪ (B − A)
- 4 → d: f(x) = x² is not one-one (f(1) = f(−1) = 1) and not onto (negative numbers have no pre-image in R)
- 5 → e: f(x) = 3x + 1 is linear with non-zero slope — both one-one and onto, hence bijective
- 6 → f: The defining property of a reflexive relation
7. V. Answer Briefly (2-mark questions)
Q1. Write the set A = {x : x is a positive integer and x² < 20} in roster form.
Q2. If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, find A − B and A ∩ B.
Q3. If A = {a, b} and B = {1, 2, 3}, find A × B and n(A × B).
Q4. Check whether the relation R = {(1,1), (1,2), (2,1)} on A = {1, 2} is symmetric.
Check transitivity: (1,2) and (2,1) ∈ R, so (1,1) must ∈ R. Yes, (1,1) ∈ R. Also (2,1) and (1,2) ∈ R, so (2,2) must ∈ R. But (2,2) ∉ R. So R is not transitive.
8. VI. Answer in Detail (5-mark questions)
Q1. In a group of 400 students, 250 can speak Tamil, 200 can speak English, and 100 can speak both. Find how many can speak either Tamil or English and how many can speak only Tamil.
Worked Solution:
Q2. Let A = {1, 2, 3, 4, 5, 6} and R be a relation on A defined by R = {(a, b) : a + b ≤ 5}. Find R, its domain, range, and check if R is reflexive, symmetric, or transitive.
Worked Solution:
Q3. Let f(x) = x² and g(x) = 2x + 3 be two functions from R → R. Find (f ∘ g)(x), (g ∘ f)(x), and determine whether each composition is one-one or onto.
Worked Solution:
Q4. Verify De Morgan's Laws using Venn diagram reasoning for A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8}.
Worked Solution:
9. Common Mistakes to Avoid
| # | Mistake | Correct Approach | ||
|---|---|---|---|---|
| 1 | Writing A × B = B × A | A × B ≠ B × A in general; ordered pairs are directional | ||
| 2 | Saying n(P(A)) = n(A) | n(P(A)) = 2ⁿ where n = \ | A\ | ; e.g., n(A)=3 gives 8 subsets, not 3 |
| 3 | Confusing A ⊂ B with A ∈ B | ⊂ means "is a subset of" (every element); ∈ means "is an element of" | ||
| 4 | Forgetting the empty set when listing subsets | ∅ is always a subset; a set with n elements has 2ⁿ subsets total | ||
| 5 | Saying f(x₁) = f(x₂) ⟹ x₁ = x₂ proves one-one | The correct test is: f(x₁) = f(x₂) ⟹ x₁ = x₂ — this IS correct, but many students reverse the direction | ||
| 6 | Assuming every onto function is one-one | Onto and one-one are independent properties; verify each separately | ||
| 7 | Writing A − B = B − A | They are different: A − B = {x ∈ A : x ∉ B}, B − A = {x ∈ B : x ∉ A} | ||
| 8 | Omitting (a, a) when checking reflexivity | Every element must map to itself; check all a ∈ A | ||
| 9 | Using n(A ∪ B) = n(A) + n(B) without intersection correction | Always use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) | ||
| 10 | Confusing "number of subsets" with "number of proper subsets" | Proper subsets exclude the set itself: 2ⁿ − 1 |
10. Exam Checklist
Before you walk into the exam, confirm you can do each of these:
- [ ] Convert between roster form and set-builder form quickly and accurately
- [ ] Apply the four set operation formulas: union, intersection, difference, complement
- [ ] Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) for 2-set problems
- [ ] Use the 3-set formula for n(A ∪ B ∪ C) — know all 7 terms
- [ ] Apply De Morgan's Laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′
- [ ] Compute Cartesian products and state n(A × B) = n(A) · n(B)
- [ ] Identify relation types: check reflexive, symmetric, transitive systematically
- [ ] Check one-one: assume f(a) = f(b) and show a = b (or find a counterexample)
- [ ] Check onto: for every b ∈ codomain, solve for a ∈ domain such that f(a) = b
- [ ] Compute compositions: (f ∘ g)(x) = f(g(x)) — always apply the inner function first
- [ ] Find domain and range of a relation from its ordered pairs
- [ ] Count subsets and power set elements: |P(A)| = 2ⁿ, proper subsets = 2ⁿ − 1
- [ ] Draw Venn diagrams to verify set identity answers
- [ ] Distinguish between relations and functions: every function is a relation, not vice versa