What are De Morgan's Laws for sets?

De Morgan's Laws state: (1) (A ∪ B)' = A' ∩ B' — the complement of the union equals the intersection of the complements, and (2) (A ∩ B)' = A' ∪ B' — the complement of the intersection equals the union of the complements.

What is the difference between a subset and a proper subset?

A subset A ⊆ B means every element of A is also in B (A may equal B). A proper subset A ⊂ B means every element of A is in B, but A has at least one element fewer than B (A cannot equal B). Every set is a subset of itself, but no set is a proper subset of itself.

What is the power set of a set?

The power set P(A) is the set of all subsets of A, including the empty set and A itself. If n(A) = n, then n(P(A)) = 2ⁿ. For example, if A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}} and n(P(A)) = 2² = 4.

How do you convert roster form to set-builder form?

Identify the common property shared by all elements. For example, A = {2, 4, 6, 8} in roster form becomes A = {x : x is an even natural number less than 10} in set-builder form. The colon ':' is read as 'such that'.

Samacheer Kalvi 9th Maths Set Language Exercise Solutions | KalviGavani

Q: What is the cardinality of a set?

The cardinality of a set is the number of distinct elements in the set, denoted n(A). For example, if A = {1, 2, 3, 4}, then n(A) = 4. Cardinality counts each element only once, even if it is listed multiple times.

How to Use This Page

1. Solve each exercise on your own first. Attempt every question before looking at the solution. 2. Compare your method, not just the answer. The steps matter more than the final value. 3. Mark mistakes in a notebook. Writing corrections by hand improves retention. 4. Revisit weak areas. If you struggle with a concept, go back to the Key Concepts section and re-read the definitions.

Key Concepts (Quick Recap)

Set: A well-defined collection of distinct objects. Denoted by capital letters (A, B, C...).
Element: An object in a set. x ∈ A means "x belongs to A"; x ∉ A means "x does not belong to A."
Roster (Tabular) Form: List all elements inside curly braces. Example: A = {2, 4, 6, 8}.
Set-Builder Form: Describe the common property. Example: A = {x : x is an even natural number less than 10}.
Empty Set (∅): A set with no elements. Also called null set or void set.
Singleton Set: A set with exactly one element. Example: {5}.
Finite Set: A set with a countable number of elements.
Infinite Set: A set with an uncountable number of elements. Example: N = {1, 2, 3, ...}.
Equal Sets: Two sets are equal if they contain exactly the same elements. A = B.
Equivalent Sets: Two sets have the same cardinality (same number of elements) but may contain different elements.
Subset (A ⊆ B): Every element of A is also in B.
Proper Subset (A ⊂ B): A is a subset of B but A ≠ B.
Power Set P(A): The set of all subsets of A. n(P(A)) = 2ⁿ where n = n(A).
Universal Set (U): The set containing all elements under consideration.
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'): U − A = {x : x ∈ U and x ∉ A}.
Cardinality Formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
De Morgan's Law 1: (A ∪ B)' = A' ∩ B'.
De Morgan's Law 2: (A ∩ B)' = A' ∪ B'.
n(A − B) = n(A) − n(A ∩ B).
n(A' ∩ B') = n(U) − n(A ∪ B).

Exercise 1.1 — Set Concepts, Roster Form, Set-Builder Form, Types of Sets

Question 1

Write the following sets in roster form.

(i) A = {x : x is a natural number less than 7}

Natural numbers less than 7: 1, 2, 3, 4, 5, 6
A = {1, 2, 3, 4, 5, 6}

AnswerA = {1, 2, 3, 4, 5, 6}

(ii) B = {x : x is a vowel in the English alphabet}

Vowels in English: a, e, i, o, u
B = {a, e, i, o, u}

AnswerB = {a, e, i, o, u}

(iii) C = {x : x is an integer and −3 < x < 3}

Integers strictly between −3 and 3: −2, −1, 0, 1, 2
C = {−2, −1, 0, 1, 2}

AnswerC = {−2, −1, 0, 1, 2}

(iv) D = {x : x is a prime number less than 20}

Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19
D = {2, 3, 5, 7, 11, 13, 17, 19}

AnswerD = {2, 3, 5, 7, 11, 13, 17, 19}

Question 2

Write the following sets in set-builder form.

(i) A = {3, 6, 9, 12, 15}

Each element is a multiple of 3, and all are natural numbers ≤ 15 that are multiples of 3.
A = {x : x = 3k, k ∈ N and 1 ≤ k ≤ 5}
Or equivalently: A = {x : x is a natural number, x is divisible by 3, x ≤ 15}

AnswerA = {x : x = 3k, k ∈ N, 1 ≤ k ≤ 5}

(ii) B = {1, 4, 9, 16, 25}

Each element is a perfect square: 1², 2², 3², 4², 5².
B = {x : x = k², k ∈ N, 1 ≤ k ≤ 5}

AnswerB = {x : x = k², k ∈ N, 1 ≤ k ≤ 5}

(iii) C = {a, e, i, o, u}

These are the vowels of the English alphabet.
C = {x : x is a vowel in the English alphabet}

AnswerC = {x : x is a vowel in the English alphabet}

(iv) D = {2, 4, 6, 8, ..., 50}

Even natural numbers from 2 to 50.
D = {x : x is an even natural number, 2 ≤ x ≤ 50}

AnswerD = {x : x is an even natural number, 2 ≤ x ≤ 50}

Question 3

Which of the following are empty sets?

(i) A = {x : x is a natural number less than 1}

Natural numbers start from 1. There is no natural number less than 1.
Therefore, A has no elements.
A is an empty set.

AnswerA = ∅ (empty set)

(ii) B = {x : x is an even prime number greater than 2}

The only even prime number is 2. No even prime number is greater than 2.
Therefore, B has no elements.
B is an empty set.

AnswerB = ∅ (empty set)

(iii) C = {x : x² = 4 and x is an odd integer}

x² = 4 gives x = 2 or x = −2. Both are even integers.
No odd integer satisfies x² = 4.
Therefore, C has no elements.
C is an empty set.

AnswerC = ∅ (empty set)

(iv) D = {x : x is a month with 32 days}

No month has 32 days.
Therefore, D has no elements.
D is an empty set.

AnswerD = ∅ (empty set)

Question 4

Which of the following are singleton sets?

(i) A = {x : x + 3 = 5, x ∈ N}

x + 3 = 5 ⟹ x = 2
Only one element: {2}
A is a singleton set.

AnswerYes, A = {2} is a singleton set.

(ii) B = {x : x is an even prime number}

The only even prime number is 2.
B = {2}
B is a singleton set.

AnswerYes, B = {2} is a singleton set.

(iii) C = {x : x² − 9 = 0, x ∈ N}

x² − 9 = 0 ⟹ x² = 9 ⟹ x = 3 (since x ∈ N, we ignore x = −3)
C = {3}
C is a singleton set.

AnswerYes, C = {3} is a singleton set.

Question 5

State whether the following pairs of sets are equal or equivalent.

(i) A = {1, 2, 3} and B = {a, b, c}

A has elements 1, 2, 3. B has elements a, b, c.
Both sets have 3 elements (n(A) = n(B) = 3).
But the elements are different, so A ≠ B.
The sets are equivalent (same cardinality) but not equal.

AnswerEquivalent but not equal.

(ii) A = {1, 2, 3} and B = {3, 2, 1}

A = {1, 2, 3} and B = {3, 2, 1}.
Both contain exactly the same elements (order does not matter in sets).
A = B. The sets are equal.

AnswerEqual (and therefore also equivalent).

(iii) A = {x : x is a letter in the word "MATH"} and B = {M, A, T, H}

A = {M, A, T, H} (from set-builder form)
B = {M, A, T, H}
Same elements ⟹ A = B.

AnswerEqual.

Question 6

Classify the following as finite or infinite sets.

(i) A = {x : x is a natural number}

Natural numbers: {1, 2, 3, ...} — goes on forever.
A is an infinite set.

AnswerInfinite set.

(ii) B = {x : x is a prime number less than 100}

Prime numbers less than 100 are countable and limited.
B = {2, 3, 5, 7, 11, ..., 97} — there are exactly 25 such primes.
B is a finite set.

AnswerFinite set.

(iii) C = {x : x is a day of the week}

There are exactly 7 days.
C = {Monday, Tuesday, ..., Sunday}.
C is a finite set with n(C) = 7.

AnswerFinite set.

(iv) D = {x : x is a point on a line}

A line has infinitely many points.
D is an infinite set.

AnswerInfinite set.

Question 7

Find the cardinality of each set.

(i) A = {a, e, i, o, u}

Elements: a, e, i, o, u
Number of elements: 5
n(A) = 5

Answern(A) = 5

(ii) B = {x : x is a prime factor of 30}

Prime factorisation of 30: 2 × 3 × 5
Prime factors: {2, 3, 5}
n(B) = 3

Answern(B) = 3

(iii) C = {x : x² − 5x + 6 = 0}

x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x = 2 or x = 3
C = {2, 3}
n(C) = 2

Answern(C) = 2

(iv) D = {x : x is a letter in the word "CALCULUS"}

Letters in CALCULUS: C, A, L, C, U, L, U, S
Distinct letters: {C, A, L, U, S}
n(D) = 5

Answern(D) = 5

Question 8

Write the set of all integers between −4 and 4 in roster form. Also write its cardinality.

Integers between −4 and 4 (exclusive): −3, −2, −1, 0, 1, 2, 3
A = {−3, −2, −1, 0, 1, 2, 3}
n(A) = 7

AnswerA = {−3, −2, −1, 0, 1, 2, 3}, n(A) = 7

Question 9

Which of the following collections are sets? Give reasons.

(i) The collection of all good students in a class.

"Good" is not well-defined — different people may have different opinions.
This is NOT a set.

AnswerNot a set — "good" is not a well-defined criterion.

(ii) The collection of all even numbers between 1 and 20.

"Even numbers between 1 and 20" is well-defined — we can definitely determine
whether any number belongs or not.
This IS a set: {2, 4, 6, 8, 10, 12, 14, 16, 18}

AnswerYes, this is a set — the criterion is well-defined.

(iii) The collection of the five most talented cricketers of India.

"Most talented" is subjective — different people may choose different players.
This is NOT a set.

AnswerNot a set — "most talented" is subjective and not well-defined.

Question 10

If A = {1, {2}, {3, 4}}, find n(A) and list all elements.

A has 3 elements:
  Element 1: 1 (a number)
  Element 2: {2} (a set containing 2)
  Element 3: {3, 4} (a set containing 3 and 4)
Note: {2} and {3, 4} are single elements of A, even though they are sets themselves.
n(A) = 3

Answern(A) = 3. Elements are 1, {2}, {3, 4}.

Question 11

Let A = {x : x is a letter in the word "TAMILNADU"}. Write A in roster form and find n(A).

Letters in TAMILNADU: T, A, M, I, L, N, A, D, U
Distinct letters: {T, A, M, I, L, N, D, U}
(Note: A appears twice, but in a set it is listed only once.)
A = {T, A, M, I, L, N, D, U}
n(A) = 8

AnswerA = {T, A, M, I, L, N, D, U}, n(A) = 8

Question 12

If A = {1, 2, 3, 4, 5} and B = {x : x = 2k − 1, k ∈ N, k ≤ 3}, are A and B equal?

B: x = 2k − 1
  k = 1 ⟹ x = 1
  k = 2 ⟹ x = 3
  k = 3 ⟹ x = 5
B = {1, 3, 5}

A = {1, 2, 3, 4, 5}
B = {1, 3, 5}
A has 5 elements, B has 3 elements.
Since B has elements that A has, but A has elements that B doesn't (2 and 4), A ≠ B.

AnswerA ≠ B. They are neither equal nor equivalent.

Question 13

Write two different set-builder descriptions for the set A = {2, 4, 6, 8, 10}.

Description 1: A = {x : x is an even natural number less than or equal to 10}
Description 2: A = {x : x = 2k, k ∈ N, 1 ≤ k ≤ 5}
Both describe the same set {2, 4, 6, 8, 10}.

Answer(1) A = {x : x is an even natural number, x ≤ 10}; (2) A = {x : x = 2k, k ∈ N, 1 ≤ k ≤ 5}

Question 14

Insert the appropriate symbol ∈ or ∉.

(i) 5 __ {1, 3, 5, 7}

5 is an element of {1, 3, 5, 7}.
5 ∈ {1, 3, 5, 7}

Answer5 ∈ {1, 3, 5, 7}

(ii) 4 __ {1, 3, 5, 7}

4 is not an element of {1, 3, 5, 7}.
4 ∉ {1, 3, 5, 7}

Answer4 ∉ {1, 3, 5, 7}

(iii) {3} __ {1, 2, {3}, 4}

{3} is an element of {1, 2, {3}, 4} because {3} itself appears in the set.
{3} ∈ {1, 2, {3}, 4}

Answer{3} ∈ {1, 2, {3}, 4}

Question 15

Which of the following sets are empty?

(i) A = {x : x is a natural number between 5 and 6}

There is no natural number between 5 and 6.
A = ∅ (empty set)

AnswerA is an empty set.

(ii) B = {x : x is an integer satisfying x² = 9 and x < −5}

x² = 9 ⟹ x = 3 or x = −3
Neither 3 nor −3 is less than −5.
B = ∅ (empty set)

AnswerB is an empty set.

(iii) C = {x : x ∈ N and 2x + 1 = 0}

2x + 1 = 0 ⟹ x = −1/2
−1/2 is not a natural number.
C = ∅ (empty set)

AnswerC is an empty set.

Question 16

Write the set A = {x : 2x + 3 = 11, x ∈ N} in roster form.

2x + 3 = 11
2x = 11 − 3 = 8
x = 4
Since x ∈ N and x = 4, A = {4}

AnswerA = {4}

Question 17

Write A = {x : x is a day of the week starting with 'S'} in roster form.

Days starting with S: Saturday, Sunday
A = {Saturday, Sunday}

AnswerA = {Saturday, Sunday}

Question 18

If A = {x : x is a prime number, x < 10} and B = {x : x is an odd number less than 10}, find A ∩ B.

A = {2, 3, 5, 7} (primes less than 10)
B = {1, 3, 5, 7, 9} (odd numbers less than 10)
A ∩ B = {3, 5, 7} (elements common to both)

AnswerA ∩ B = {3, 5, 7}

Question 19

Determine whether the following sets are equal: A = {1, 2, 3} and B = {3, 1, 2}. Justify your answer.

A = {1, 2, 3}
B = {3, 1, 2}
Both sets contain the same elements: 1, 2, and 3.
In a set, the order of elements does not matter.
Therefore A = B.

AnswerA = B. The order of elements in a set does not matter; both sets contain exactly the same elements.

Question 20

Express the set A = {x : x ∈ Z, −2 ≤ x ≤ 2} in roster form.

Integers from −2 to 2 (inclusive): −2, −1, 0, 1, 2
A = {−2, −1, 0, 1, 2}

AnswerA = {−2, −1, 0, 1, 2}

Exercise 1.2 — Subset, Proper Subset, Power Set, Universal Set

Question 1

Write all the subsets of A = {a, b}.

Step 1: n(A) = 2, so number of subsets = 2² = 4
Step 2: List all subsets:
  Subset with 0 elements: ∅
  Subset with 1 element: {a}, {b}
  Subset with 2 elements: {a, b}
All subsets: ∅, {a}, {b}, {a, b}

AnswerSubsets of A: ∅, {a}, {b}, {a, b}

Question 2

Write all the subsets of B = {1, 2, 3}. How many subsets are there?

n(B) = 3, so number of subsets = 2³ = 8
Subsets:
  0 elements: ∅
  1 element:  {1}, {2}, {3}
  2 elements: {1, 2}, {1, 3}, {2, 3}
  3 elements: {1, 2, 3}
Total = 8 subsets

Answer∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} — 8 subsets.

Question 3

Find the power set of A = {p, q}.

Power set P(A) = set of all subsets of A.
Subsets of A: ∅, {p}, {q}, {p, q}
P(A) = {∅, {p}, {q}, {p, q}}
n(P(A)) = 2² = 4

AnswerP(A) = {∅, {p}, {q}, {p, q}}

Question 4

If n(A) = 4, find n(P(A)), the number of subsets of A, and the number of proper subsets.

n(A) = 4
Number of subsets = n(P(A)) = 2⁴ = 16
Number of proper subsets = 2⁴ − 1 = 15
  (All subsets except A itself)

Answern(P(A)) = 16. Proper subsets = 15.

Question 5

If A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6}, determine whether B is a subset and/or proper subset of A.

Every element of B = {2, 4, 6} is in A = {1, 2, 3, 4, 5, 6}.
  2 ∈ A? Yes. 4 ∈ A? Yes. 6 ∈ A? Yes.
Therefore B ⊆ A.
Since B ≠ A (A has elements that B doesn't), B is also a proper subset.
B ⊂ A.

AnswerB ⊆ A and B ⊂ A (B is both a subset and a proper subset of A).

Question 6

Let A = {1, 2, 3}. Which of the following are true?

(i) {1} ⊆ A

{1} is a set containing 1. 1 ∈ A. So every element of {1} is in A.
{1} ⊆ A is TRUE.

AnswerTrue.

(ii) {1, 4} ⊆ A

4 ∈ {1, 4} but 4 ∉ A.
So {1, 4} ⊆ A is FALSE.

AnswerFalse.

(iii) ∅ ⊆ A

The empty set is a subset of every set.
∅ ⊆ A is TRUE.

AnswerTrue.

(iv) A ⊆ A

Every set is a subset of itself.
A ⊆ A is TRUE.

AnswerTrue.

Question 7

If A ⊂ B and B ⊂ C, can we conclude that A ⊂ C? Give an example.

Yes. If A ⊂ B, every element of A is in B (and A ≠ B).
If B ⊂ C, every element of B is in C (and B ≠ C).
Therefore, every element of A is in C, and since A ≠ B and B ≠ C, A ≠ C.
So A ⊂ C.
Example: A = {1}, B = {1, 2}, C = {1, 2, 3}
A ⊂ B and B ⊂ C ⟹ A ⊂ C

AnswerYes, A ⊂ C. Example: {1} ⊂ {1, 2} ⊂ {1, 2, 3} implies {1} ⊂ {1, 2, 3}.

Question 8

How many elements does a set have if its power set has 64 elements?

n(P(A)) = 64
2^(n(A)) = 64
2^(n(A)) = 2⁶
n(A) = 6

Answern(A) = 6

Question 9

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8}, find A' (complement of A).

A' = U − A = elements in U that are NOT in A
A' = {1, 3, 5, 7, 9, 10}

AnswerA' = {1, 3, 5, 7, 9, 10}

Question 10

If U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {2, 4, 6}, verify that A ∪ B and A ∩ B are subsets of U.

A ∪ B = {1, 2, 3, 4, 6}
Every element of A ∪ B is in U? Check: 1✓, 2✓, 3✓, 4✓, 6✓. Yes.
A ∪ B ⊆ U is TRUE.

A ∩ B = {2}
Every element of A ∩ B is in U? Check: 2✓. Yes.
A ∩ B ⊆ U is TRUE.

AnswerBoth A ∪ B ⊆ U and A ∩ B ⊆ U are verified.

Question 11

Write all the proper subsets of C = {1, 2, 3, 4}.

Proper subsets = all subsets except C itself.
n(C) = 4, total subsets = 2⁴ = 16, proper subsets = 16 − 1 = 15
  0 elements: ∅
  1 element:  {1}, {2}, {3}, {4}
  2 elements: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}
  3 elements: {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}
Total = 15 proper subsets

Answer∅, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} — 15 proper subsets.

Question 12

If A = {x : x ∈ N, x ≤ 5} and B = {x : x ∈ N, x is odd and x ≤ 5}, show that B ⊂ A.

A = {1, 2, 3, 4, 5}
B = {1, 3, 5}
Every element of B = {1, 3, 5} is in A = {1, 2, 3, 4, 5}: Yes.
B ≠ A (A has elements 2 and 4 that B doesn't have).
Therefore B ⊂ A.

AnswerB = {1, 3, 5} ⊂ {1, 2, 3, 4, 5} = A. Verified.

Question 13

Find the number of subsets and the number of proper subsets of a set with 6 elements.

n(A) = 6
Number of subsets = 2⁶ = 64
Number of proper subsets = 2⁶ − 1 = 63

Answer64 subsets, 63 proper subsets.

Question 14

Is ∅ ⊆ {1, 2, 3}? Is ∅ ∈ {1, 2, 3}?

∅ ⊆ {1, 2, 3}: TRUE. The empty set is a subset of every set.
∅ ∈ {1, 2, 3}: FALSE. ∅ is not an element of {1, 2, 3}.
  The elements of {1, 2, 3} are 1, 2, and 3 — not ∅.

Answer∅ ⊆ {1, 2, 3} is TRUE. ∅ ∈ {1, 2, 3} is FALSE.

Question 15

If A = {1, 2} and B = {{1}, {2}, {1, 2}, ∅}, what is the relationship between A and B?

B is the power set of A.
P(A) = {∅, {1}, {2}, {1, 2}} = {∅, {1}, {2}, {1, 2}}
This matches B exactly.
B = P(A)

AnswerB = P(A). B is the power set of A.

Question 16

Let U = {1, 2, 3, 4, ..., 10}. A = {1, 3, 5, 7, 9}. Find A' and n(A').

A' = U − A = {2, 4, 6, 8, 10}
n(A') = 5

AnswerA' = {2, 4, 6, 8, 10}, n(A') = 5

Question 17

If A ⊂ B, what can you say about A ∪ B?

Since A ⊂ B, every element of A is already in B.
A ∪ B contains all elements of B plus any elements of A not in B.
But since A ⊂ B, all elements of A are already in B.
Therefore A ∪ B = B.

AnswerA ∪ B = B (because every element of A is already in B).

Question 18

If A ⊂ B, what can you say about A ∩ B?

Since A ⊂ B, every element of A is in B.
A ∩ B = elements common to A and B = all elements of A (since every element of A is in B).
Therefore A ∩ B = A.

AnswerA ∩ B = A (because every element of A is in B).

Exercise 1.3 — Set Operations: Union, Intersection, Difference, Complement

Question 1

If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, find A ∪ B.

A ∪ B = {x : x ∈ A or x ∈ B}
Elements in A: 1, 2, 3, 4, 5
Elements in B: 3, 4, 5, 6, 7
Combined (no duplicates): 1, 2, 3, 4, 5, 6, 7
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

AnswerA ∪ B = {1, 2, 3, 4, 5, 6, 7}

Question 2

If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, find A ∩ B.

A ∩ B = {x : x ∈ A and x ∈ B}
Common elements: 3, 4, 5
A ∩ B = {3, 4, 5}

AnswerA ∩ B = {3, 4, 5}

Question 3

If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, find A − B and B − A.

A − B = {x : x ∈ A and x ∉ B}
  Elements in A but NOT in B: 1, 2
  A − B = {1, 2}

B − A = {x : x ∈ B and x ∉ A}
  Elements in B but NOT in A: 6, 7
  B − A = {6, 7}

AnswerA − B = {1, 2}; B − A = {6, 7}

Question 4

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, find A'.

A' = U − A = {x : x ∈ U and x ∉ A}
A' = {6, 7, 8, 9, 10}

AnswerA' = {6, 7, 8, 9, 10}

Question 5

If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {3, 4, 5}, find (A ∪ B)'.

Step 1: A ∪ B = {1, 2, 3, 4, 5}
Step 2: (A ∪ B)' = U − (A ∪ B) = {6, 7, 8}

Answer(A ∪ B)' = {6, 7, 8}

Question 6

If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {3, 4, 5}, find A' ∩ B'.

A' = U − A = {4, 5, 6, 7, 8}
B' = U − B = {1, 2, 6, 7, 8}
A' ∩ B' = common elements of A' and B' = {6, 7, 8}

AnswerA' ∩ B' = {6, 7, 8}

Note: (A ∪ B)' = {6, 7, 8} = A' ∩ B'. This verifies De Morgan's First Law!

Question 7

If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {3, 4, 5}, find (A ∩ B)' and A' ∪ B'.

Step 1: A ∩ B = {3} (only common element)
Step 2: (A ∩ B)' = U − {3} = {1, 2, 4, 5, 6, 7, 8}

Step 3: A' = {4, 5, 6, 7, 8}
Step 4: B' = {1, 2, 6, 7, 8}
Step 5: A' ∪ B' = {1, 2, 4, 5, 6, 7, 8}

Answer(A ∩ B)' = {1, 2, 4, 5, 6, 7, 8}; A' ∪ B' = {1, 2, 4, 5, 6, 7, 8}

Note: (A ∩ B)' = A' ∪ B'. This verifies De Morgan's Second Law!

Question 8

If A = {a, b, c, d, e}, B = {c, d, e, f, g}, find A ∪ B and A ∩ B.

A ∪ B = {a, b, c, d, e, f, g}
A ∩ B = {c, d, e}

AnswerA ∪ B = {a, b, c, d, e, f, g}; A ∩ B = {c, d, e}

Question 9

If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}, find A ∩ B and A ∪ B.

A ∩ B = {x : x ∈ A and x ∈ B}
No elements are common (A has only odd, B has only even).
A ∩ B = ∅ (empty set)

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

AnswerA ∩ B = ∅; A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Question 10

Verify De Morgan's Law (A ∪ B)' = A' ∩ B' for U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, B = {2, 3, 4}.

LHS: (A ∪ B)' = ?
  A ∪ B = {1, 2, 3, 4, 5}
  (A ∪ B)' = U − (A ∪ B) = {6}

RHS: A' ∩ B' = ?
  A' = {2, 4, 6}
  B' = {1, 5, 6}
  A' ∩ B' = {6}

LHS = RHS = {6}
De Morgan's First Law verified!

AnswerBoth sides equal {6}. De Morgan's Law (A ∪ B)' = A' ∩ B' verified.

Question 11

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, find A ∩ (B ∩ C).

Step 1: B ∩ C = {5, 6} (common elements of B and C)
Step 2: A ∩ {5, 6} = ∅ (no common elements between A and {5, 6})
A ∩ (B ∩ C) = ∅

AnswerA ∩ (B ∩ C) = ∅

Question 12

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, find A ∪ (B ∪ C).

Step 1: B ∪ C = {3, 4, 5, 6, 7, 8}
Step 2: A ∪ {3, 4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
A ∪ (B ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8}

AnswerA ∪ (B ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8}

Question 13

If A ⊂ B, what is A ∩ B? What is A ∪ B?

Since A ⊂ B, every element of A is already in B.
A ∩ B = {x : x ∈ A and x ∈ B} = {x : x ∈ A} = A
A ∪ B = {x : x ∈ A or x ∈ B} = B

AnswerIf A ⊂ B, then A ∩ B = A and A ∪ B = B.

Question 14

If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, find A − B and B − A.

A − B = {x : x ∈ A and x ∉ B}
  Elements in A: 1, 2, 3, 4, 5, 6
  Elements in B: 2, 4, 6, 8
  Elements in A but NOT in B: 1, 3, 5
  A − B = {1, 3, 5}

B − A = {x : x ∈ B and x ∉ A}
  Elements in B: 2, 4, 6, 8
  Elements in A: 1, 2, 3, 4, 5, 6
  Elements in B but NOT in A: 8
  B − A = {8}

AnswerA − B = {1, 3, 5}; B − A = {8}

Question 15

If U = {1, 2, 3, ..., 10}, A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}, find A' and B'. What do you notice?

A' = U − A = {1, 3, 5, 7, 9} = B
B' = U − B = {2, 4, 6, 8, 10} = A

Observation: A' = B and B' = A.
This is because A and B partition U into even and odd numbers — they are complements of each other.

AnswerA' = {1, 3, 5, 7, 9} = B; B' = {2, 4, 6, 8, 10} = A. A and B are complements of each other.

Question 16

Using Venn diagram regions, express the following in terms of A, B, and U: (A ∪ B) − (A ∩ B). What is this set called?

A ∪ B = all elements in A or B
A ∩ B = elements in both A and B
(A ∪ B) − (A ∩ B) = elements in A or B but NOT in both
This is called the **symmetric difference** of A and B, denoted A Δ B.
It equals (A − B) ∪ (B − A).

Answer(A ∪ B) − (A ∩ B) = (A − B) ∪ (B − A). This is called the symmetric difference A Δ B.

Question 17

If A and B are disjoint sets with n(A) = 5 and n(B) = 7, find n(A ∪ B).

Since A and B are disjoint, A ∩ B = ∅, so n(A ∩ B) = 0.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
          = 5 + 7 − 0
          = 12

Answern(A ∪ B) = 12

Question 18

If A = {x : x is an even natural number less than 10} and B = {x : x is a multiple of 3 less than 10}, find A ∪ B and A ∩ B.

A = {2, 4, 6, 8}
B = {3, 6, 9}
A ∪ B = {2, 3, 4, 6, 8, 9}
A ∩ B = {6} (6 is the only even multiple of 3 less than 10)

AnswerA ∪ B = {2, 3, 4, 6, 8, 9}; A ∩ B = {6}

Exercise 1.4 — Cardinality, De Morgan's Laws, Venn Diagrams, Word Problems

Question 1

If n(A) = 12, n(B) = 8, and n(A ∩ B) = 3, find n(A ∪ B).

Using the cardinality formula:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
         = 12 + 8 − 3
         = 20 − 3
         = 17

Answern(A ∪ B) = 17

Question 2

If n(A) = 10, n(B) = 15, and n(A ∪ B) = 20, find n(A ∩ B).

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
20 = 10 + 15 − n(A ∩ B)
20 = 25 − n(A ∩ B)
n(A ∩ B) = 25 − 20
n(A ∩ B) = 5

Answern(A ∩ B) = 5

Question 3

In a class of 50 students, 30 like Mathematics and 25 like Science. If every student likes at least one subject, how many like both?

Let M = students who like Mathematics, S = students who like Science.
n(M) = 30, n(S) = 25, n(M ∪ S) = 50 (every student likes at least one)
n(M ∪ S) = n(M) + n(S) − n(M ∩ S)
50 = 30 + 25 − n(M ∩ S)
50 = 55 − n(M ∩ S)
n(M ∩ S) = 55 − 50
n(M ∩ S) = 5

Answer5 students like both Mathematics and Science.

Question 4

In a group of 100 people, 60 can speak Tamil, 40 can speak English, and 20 can speak both. How many can speak at least one language? How many can speak only Tamil?

Let T = Tamil speakers, E = English speakers.
n(T) = 60, n(E) = 40, n(T ∩ E) = 20

At least one language = n(T ∪ E)
n(T ∪ E) = n(T) + n(E) − n(T ∩ E)
          = 60 + 40 − 20
          = 80

Only Tamil = n(T) − n(T ∩ E)
           = 60 − 20
           = 40

Answer80 people can speak at least one language. 40 people can speak only Tamil.

Question 5

In a class of 40 students, 25 play cricket, 20 play football, and 10 play both. How many students play neither?

Let C = cricket players, F = football players.
n(C) = 25, n(F) = 20, n(C ∩ F) = 10

Students who play at least one = n(C ∪ F)
n(C ∪ F) = n(C) + n(F) − n(C ∩ F)
         = 25 + 20 − 10
         = 35

Students who play neither = 40 − 35 = 5

Answer5 students play neither cricket nor football.

Question 6

If n(A) = 20, n(B) = 15, n(A ∪ B) = 30, find n(A ∩ B) and n(A − B).

Step 1: Find n(A ∩ B)
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
30 = 20 + 15 − n(A ∩ B)
30 = 35 − n(A ∩ B)
n(A ∩ B) = 35 − 30 = 5

Step 2: Find n(A − B)
n(A − B) = n(A) − n(A ∩ B)
         = 20 − 5
         = 15

Answern(A ∩ B) = 5, n(A − B) = 15

Question 7

Verify De Morgan's Law (A ∪ B)' = A' ∩ B' for U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}.

LHS: (A ∪ B)'
  A ∪ B = {1, 2, 3, 4, 5, 6}
  (A ∪ B)' = {7, 8}

RHS: A' ∩ B'
  A' = {5, 6, 7, 8}
  B' = {1, 2, 7, 8}
  A' ∩ B' = {7, 8}

LHS = RHS = {7, 8}
De Morgan's First Law verified!

Answer(A ∪ B)' = A' ∩ B' = {7, 8}. Verified.

Question 8

Verify De Morgan's Law (A ∩ B)' = A' ∪ B' using the same sets.

U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}

LHS: (A ∩ B)'
  A ∩ B = {3, 4}
  (A ∩ B)' = {1, 2, 5, 6, 7, 8}

RHS: A' ∪ B'
  A' = {5, 6, 7, 8}
  B' = {1, 2, 7, 8}
  A' ∪ B' = {1, 2, 5, 6, 7, 8}

LHS = RHS = {1, 2, 5, 6, 7, 8}
De Morgan's Second Law verified!

Answer(A ∩ B)' = A' ∪ B' = {1, 2, 5, 6, 7, 8}. Verified.

Question 9

In a survey of 200 people, it was found that 120 like tea, 90 like coffee, and 50 like both. Find: (i) How many like at least one drink? (ii) How many like only tea? (iii) How many like only coffee? (iv) How many like neither?

Let T = tea lovers, C = coffee lovers.
n(T) = 120, n(C) = 90, n(T ∩ C) = 50

(i) At least one = n(T ∪ C)
    n(T ∪ C) = n(T) + n(C) − n(T ∩ C)
             = 120 + 90 − 50
             = 160

(ii) Only tea = n(T) − n(T ∩ C) = 120 − 50 = 70

(iii) Only coffee = n(C) − n(T ∩ C) = 90 − 50 = 40

(iv) Neither = 200 − n(T ∪ C) = 200 − 160 = 40

Answer(i) 160, (ii) 70, (iii) 40, (iv) 40

Question 10

If n(U) = 50, n(A) = 30, n(B) = 25, n(A ∪ B) = 45, find n(A ∩ B), n(A − B), n(B − A), and n((A ∪ B)').

n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
         = 30 + 25 − 45
         = 10

n(A − B) = n(A) − n(A ∩ B) = 30 − 10 = 20

n(B − A) = n(B) − n(A ∩ B) = 25 − 10 = 15

n((A ∪ B)') = n(U) − n(A ∪ B) = 50 − 45 = 5

Answern(A ∩ B) = 10, n(A − B) = 20, n(B − A) = 15, n((A ∪ B)') = 5

Question 11

In a school, there are 150 students. 80 study Physics, 70 study Chemistry, and 30 study both. Draw a Venn diagram and find how many study neither subject.

Venn diagram regions:
  Only Physics = 80 − 30 = 50
  Only Chemistry = 70 − 30 = 40
  Both = 30

Total studying at least one = 50 + 40 + 30 = 120
  (or: n(P ∪ C) = 80 + 70 − 30 = 120)

Neither = 150 − 120 = 30

Answer30 students study neither Physics nor Chemistry.

Question 12

If n(A − B) = 15, n(A ∩ B) = 10, n(B − A) = 8, find n(A), n(B), n(A ∪ B), and n(U) if n((A ∪ B)') = 12.

n(A) = n(A − B) + n(A ∩ B) = 15 + 10 = 25
n(B) = n(B − A) + n(A ∩ B) = 8 + 10 = 18
n(A ∪ B) = n(A − B) + n(A ∩ B) + n(B − A)
         = 15 + 10 + 8
         = 33
n(U) = n(A ∪ B) + n((A ∪ B)') = 33 + 12 = 45

Answern(A) = 25, n(B) = 18, n(A ∪ B) = 33, n(U) = 45

Question 13

In a town, 60% of the people speak Tamil, 40% speak Hindi, and 20% speak both. What percentage speaks at least one language?

n(T) = 60%, n(H) = 40%, n(T ∩ H) = 20%
n(T ∪ H) = n(T) + n(H) − n(T ∩ H)
         = 60 + 40 − 20
         = 80%

Answer80% of the people speak at least one language.

Question 14

If A and B are two sets such that n(A) = 18, n(B) = 25, and n(A ∪ B) = 40, find n(A ∩ B) and n(A − B).

n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
         = 18 + 25 − 40
         = 3

n(A − B) = n(A) − n(A ∩ B) = 18 − 3 = 15

Answern(A ∩ B) = 3, n(A − B) = 15

Question 15

In a class of 60 students, 23 play hockey, 27 play basketball, and 9 play both. How many students play only hockey? How many play only basketball? How many play neither?

Only hockey = n(H) − n(H ∩ B) = 23 − 9 = 14
Only basketball = n(B) − n(H ∩ B) = 27 − 9 = 18
At least one = n(H ∪ B) = 23 + 27 − 9 = 41
Neither = 60 − 41 = 19

AnswerOnly hockey = 14, only basketball = 18, neither = 19

Question 16

If n(U) = 100, n(A) = 50, n(B) = 40, n(A ∩ B) = 20, find n(A' ∩ B') using De Morgan's Law.

By De Morgan's Law: A' ∩ B' = (A ∪ B)'
So n(A' ∩ B') = n((A ∪ B)')

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
          = 50 + 40 − 20
          = 70

n(A' ∩ B') = n(U) − n(A ∪ B)
           = 100 − 70
           = 30

Answern(A' ∩ B') = 30

Question 17

Among 80 students, 50 scored above 80 in Maths, 30 scored above 80 in Science, and 10 scored above 80 in both. How many scored above 80 in at least one subject? How many scored above 80 in exactly one subject?

At least one = n(M ∪ S) = 50 + 30 − 10 = 70
Exactly one = (n(M) − n(M ∩ S)) + (n(S) − n(M ∩ S))
           = (50 − 10) + (30 − 10)
           = 40 + 20
           = 60

Answer70 students scored above 80 in at least one subject; 60 scored above 80 in exactly one subject.

Question 18

If A ∪ B = A ∩ B, prove that A = B.

Given: A ∪ B = A ∩ B

We know that A ∩ B ⊆ A ⊆ A ∪ B (always true for any two sets).
If A ∪ B = A ∩ B, then:
  A ∩ B ⊆ A ⊆ A ∪ B = A ∩ B
This means A ⊆ A ∩ B and A ∩ B ⊆ A, so A = A ∩ B.
Since A = A ∩ B, every element of A is in B (from the definition of intersection).
So A ⊆ B.

Similarly, A ∩ B ⊆ B ⊆ A ∪ B = A ∩ B.
This means B ⊆ A ∩ B and A ∩ B ⊆ B, so B = A ∩ B.
Since B = A ∩ B, every element of B is in A.
So B ⊆ A.

Since A ⊆ B and B ⊆ A, we have A = B.

AnswerProved. If A ∪ B = A ∩ B, then A = B.

Question 19

If n(A) = 12, n(B) = 8, and n(A ∪ B) = 15, find n(A ∩ B), n(A − B), and n(B − A).

n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
         = 12 + 8 − 15
         = 5

n(A − B) = n(A) − n(A ∩ B) = 12 − 5 = 7

n(B − A) = n(B) − n(A ∩ B) = 8 − 5 = 3

Answern(A ∩ B) = 5, n(A − B) = 7, n(B − A) = 3

Question 20

In a group of 120 students, 70 play cricket, 60 play football, and 15 play neither. Find: (i) How many play both? (ii) How many play only cricket? (iii) How many play only football?

Students who play at least one = 120 − 15 = 105
n(C ∪ F) = 105

n(C ∪ F) = n(C) + n(F) − n(C ∩ F)
105 = 70 + 60 − n(C ∩ F)
105 = 130 − n(C ∩ F)

(i) n(C ∩ F) = 130 − 105 = 25

(ii) Only cricket = n(C) − n(C ∩ F) = 70 − 25 = 45

(iii) Only football = n(F) − n(C ∩ F) = 60 − 25 = 35

Answer(i) 25 play both, (ii) 45 play only cricket, (iii) 35 play only football.

Common Mistakes

1. Counting repeated elements in cardinality. If A = {1, 2, 2, 3}, then n(A) = 3, not 4. A set contains only distinct elements — duplicates are automatically removed.

2. Confusing ∈ and ⊆. The symbol ∈ means "is an element of" and refers to membership. The symbol ⊆ means "is a subset of" and refers to containment. Example: 2 ∈ {1, 2, 3} is true, but {2} ⊆ {1, 2, 3} is true, while {2} ∈ {1, 2, 3} is false.

3. Writing A − B as elements of B that are not in A. A − B means elements of A that are NOT in B, not the other way around. A − B ≠ B − A in general.

4. Forgetting that A ∩ B can be empty. When two sets share no elements (disjoint sets), A ∩ B = ∅. This does not mean the operation failed — it is a valid result.

5. Applying n(A ∪ B) = n(A) + n(B) without subtracting the intersection. The correct formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The simple addition only works when A and B are disjoint (n(A ∩ B) = 0).

6. Assuming A' means "not in A" without specifying the universal set. The complement A' is always defined relative to the universal set U: A' = U − A. Without knowing U, you cannot determine A'.

7. Treating ∅ and {∅} as the same set. ∅ has zero elements. {∅} has one element (the empty set itself). n(∅) = 0, n({∅}) = 1.

Exam Checklist

[ ] I can write a set in roster form from a description (e.g., {x : x is even, x < 10} → {2, 4, 6, 8}).
[ ] I can convert roster form to set-builder form and vice versa.
[ ] I can identify empty, singleton, finite, infinite, equal, and equivalent sets.
[ ] I can list all subsets of a given set and know the formula n(P(A)) = 2^(n(A)).
[ ] I can distinguish between subset (⊆) and proper subset (⊂), and between ∈ and ⊆.
[ ] I can find A ∪ B, A ∩ B, A − B, and B − A for given sets.
[ ] I can find the complement A' given the universal set U.
[ ] I can verify De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
[ ] I can apply the cardinality formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) to word problems.
[ ] I can solve Venn diagram word problems — finding "only A", "only B", "both", "neither".
[ ] I can compute n(A − B) = n(A) − n(A ∩ B) and n(B − A) = n(B) − n(A ∩ B).
[ ] I remember that ∅ is a subset of every set and every set is a subset of itself.

Internal Links

[Class 9 Maths Guide — TN State Board](/class/9/maths/guide/tamil-nadu-state-board)
[Class 9 Maths Set Language Chapter Guide](/class/9/maths/guide/set-language)
[Class 9 Maths Real Numbers Exercise Solutions](/class/9/maths/solutions/real-numbers-exercise-solutions)
[Class 9 Maths Algebra Exercise Solutions](/class/9/maths/solutions/algebra-exercise-solutions)
[Class 9 Maths Geometry Exercise Solutions](/class/9/maths/solutions/geometry-exercise-solutions)
[Class 9 Maths Guide — All Chapters](/class/9/maths/guide/tamil-nadu-state-board)
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Frequently Asked Questions

The cardinality of a set is the number of distinct elements in the set, denoted n(A). For example, if A = {1, 2, 3, 4}, then n(A) = 4. Cardinality counts each element only once, even if it is listed multiple times.