Class 11 Mathematics – Chapter 1: Sets
1. Introduction to Sets
Set: A collection of well-defined and distinct objects is called a set. The objects in a set are called elements or members of the set.
Well-Defined Collection
A collection is said to be well-defined if it is clear whether a particular object belongs to the collection or not.
Representation of Sets
- Roster Form: All the elements of the set are listed, separated by commas, and enclosed in curly brackets. Example: A = {1, 2, 3, 4}
- Set-Builder Form: Elements are described using a property. Example: A = {x | x is a natural number less than 5}
Example:
Write the set of all vowels in the English alphabet.
Solution: A = {a, e, i, o, u} (Roster Form)
2. Types of Sets
| Type | Description | Example |
|---|---|---|
| Empty Set (∅) | No elements | A = {x | x is a natural number less than 1} |
| Finite Set | Countable elements | A = {2, 4, 6, 8} |
| Infinite Set | Uncountable elements | N = {1, 2, 3, 4, …} |
| Equal Sets | Same elements | A = {1, 2, 3}, B = {3, 2, 1} |
| Singleton Set | Only one element | A = {0} |
| Subsets | All elements of one set are in another | If A = {1, 2}, then {1} ⊆ A |
| Power Set | Set of all subsets | If A = {a, b}, P(A) = {∅, {a}, {b}, {a, b}} |
| Universal Set | Set containing all objects | U = {All students in school} |
Practice Set – Types of Sets
1. If A = {x | x is an even natural number less than 12}, list the set A and determine if it is finite or infinite. Justify your answer.
Answer: A = {2, 4, 6, 8, 10}. This is a finite set because the number of elements can be counted and is limited to 5.
2. Write all the subsets of the set B = {3, 5} and hence write the power set P(B). How many subsets are there in total?
Answer: Subsets of B: ∅, {3}, {5}, {3, 5}.
Power Set: P(B) = {∅, {3}, {5}, {3, 5}}. Total subsets = 22 = 4.
3. Let C = {x | x is a prime number less than 20}. Determine if C is finite or infinite. Also, list its elements.
Answer: C = {2, 3, 5, 7, 11, 13, 17, 19}. The set is finite because it has 8 countable elements.
4. A = {x | x is a multiple of 7}, is this set finite or infinite? Provide reasoning.
Answer: The set is infinite because there is no upper bound to the multiples of 7. Example: {7, 14, 21, 28, 35, …} continues indefinitely.
5. If A = {1, 2, 3} and B = {3, 4, 5}, find A ∩ B and state the type of the resulting set. Is it finite, singleton, or empty?
Answer: A ∩ B = {3}. The resulting set is a singleton set because it contains only one element.
6. Consider the universal set U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. Find Ac, the complement of A in U.
Answer: Ac = {1, 3, 5}. These are the elements in U but not in A.
7. If A = {x | x is an odd number between 10 and 20}, list the elements and determine how many subsets this set has.
Answer: A = {11, 13, 15, 17, 19}. Number of subsets = 25 = 32. The power set will have 32 elements.
8. Let A = {a, b, c} and B = {a, b, c}. Are A and B equal sets? Give reason for your answer.
Answer: Yes, A and B are equal sets because they have exactly the same elements, even though the order doesn’t matter in sets.
9. If a set P has 3 elements, how many elements will its power set contain? List all the subsets of P = {1, 2, 3}.
Answer: Power set of P has 23 = 8 elements.
Subsets: ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.
10. Is the set of rational numbers between 0 and 1 a finite or infinite set? Justify with reasoning.
Answer: The set is infinite. There are infinitely many rational numbers between any two numbers, even between 0 and 1 (e.g., 1/2, 1/3, 2/3, 1/4, etc.).
1. Which of the following is an empty set?
- A. {0}
- B. {∅}
- C. {x | x is a natural number less than 1}
- D. {1, 2, 3}
2. A set with only one element is called:
- A. Finite set
- B. Singleton set
- C. Power set
- D. Equal set
3. Which of the following sets is finite?
- A. A = {2, 4, 6, 8}
- B. A = {x | x is a whole number}
- C. A = {all stars in the universe}
- D. A = {x | x is an integer}
4. What is the power set of A = {a}?
- A. {a}
- B. {∅, {a}}
- C. {∅, a}
- D. {{a}}
5. The universal set contains:
- A. No elements
- B. All elements under consideration
- C. Only numbers
- D. Elements not in any subset
6. Which of the following is an infinite set?
- A. {2, 4, 6, 8}
- B. {a, b, c}
- C. {x | x is a natural number}
- D. {1}
7. If A = {1, 2, 3} and B = {3, 2, 1}, then:
- A. A and B are not equal
- B. A is a subset of B but not equal
- C. A and B are equal sets
- D. None of the above
8. The number of elements in the power set of A = {1, 2} is:
- A. 2
- B. 3
- C. 4
- D. 5
9. Which of the following is a subset of A = {1, 2, 3}?
- A. {4}
- B. {2}
- C. {1, 4}
- D. {0}
10. If a set has n elements, the power set has:
- A. n elements
- B. 2n elements
- C. n + 1 elements
- D. n – 1 elements
11. Which of the following is not a well-defined set?
- A. Set of odd numbers
- B. Set of red flowers
- C. Set of natural numbers
- D. Set of vowels in English
12. If A = {2, 3}, which of the following is not a subset?
- A. ∅
- B. {2}
- C. {2, 3}
- D. {1, 2}
13. A set of all students in a school is a good example of:
- A. Empty set
- B. Singleton set
- C. Finite set
- D. Universal set
14. Which of the following sets has only one element?
- A. {1, 2}
- B. ∅
- C. {0}
- D. {x | x is a vowel}
15. The set {x | x is a natural number less than 1} is:
- A. Singleton set
- B. Universal set
- C. Empty set
- D. Infinite set
3. Venn Diagrams
Venn diagrams are diagrams that show all possible logical relations between a finite collection of different sets using circles.
- Each set is represented by a circle.
- The universal set is usually represented by a rectangle.
Example:
Let U = {1,2,3,4,5,6}, A = {1,2,3}, B = {3,4,5}. Represent A and B using a Venn diagram and find A ∩ B.
Solution: A ∩ B = {3}
4. Operations on Sets
- Union (A ∪ B): Set of elements in A or B or both.
- Intersection (A ∩ B): Set of elements in both A and B.
- Difference (A − B): Elements in A but not in B.
- Complement (A′): Elements not in A, relative to the universal set.
1. A ∪ B (Union)
Example: A = {1, 2, 3}, B = {3, 4, 5} → A ∪ B = {1, 2, 3, 4, 5}
2. A ∩ B (Intersection)
Example: A = {2, 3, 4}, B = {3, 4, 5} → A ∩ B = {3, 4}
3. A′ (Complement of A)
Example: If Universal Set U = {1,2,3,4,5,6}, A = {2,4,6} → A′ = {1,3,5}
4. A − B (Difference)
Example: A = {1, 2, 3}, B = {3, 4, 5} → A − B = {1, 2}
5. A′ ∪ B′ (Complement of A ∩ B)
Example: U = {1–9}, A = {1,2,3}, B = {3,4,5} → A ∩ B = {3} → A′ ∪ B′ = U − {3} = {1,2,4,5,6,7,8,9}
Set Theory: Properties and Proofs
Properties
- Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
1. Commutative Property
Statement: A ∪ B = B ∪ A and A ∩ B = B ∩ A
Proof: By definition, union and intersection are symmetric operations. If x ∈ A ∪ B, then x ∈ A or x ∈ B. So x ∈ B ∪ A as well. Hence A ∪ B = B ∪ A.
Similarly, x ∈ A ∩ B ⇒ x ∈ A and x ∈ B ⇒ x ∈ B ∩ A.
A ∪ B = {1, 2, 3}, B ∪ A = {1, 2, 3} → Equal
A ∩ B = {2}, B ∩ A = {2} → Equal
2. Associative Property
Statement: (A ∪ B) ∪ C = A ∪ (B ∪ C)
Proof: Let x ∈ (A ∪ B) ∪ C. Then x ∈ A or x ∈ B or x ∈ C.
So x ∈ A ∪ (B ∪ C). Hence, (A ∪ B) ∪ C = A ∪ (B ∪ C).
Similar logic applies for intersection.
(A ∪ B) ∪ C = {1, 2} ∪ {2, 3} = {1, 2, 3}
A ∪ (B ∪ C) = {1} ∪ {1, 2, 3} = {1, 2, 3}
3. Distributive Property
Statement: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Proof: Let x ∈ A ∩ (B ∪ C). Then x ∈ A and (x ∈ B or x ∈ C).
So, x ∈ A ∩ B or x ∈ A ∩ C ⇒ x ∈ (A ∩ B) ∪ (A ∩ C).
A ∩ (B ∪ C) = {1, 2, 3} ∩ {2, 3} = {2, 3}
(A ∩ B) ∪ (A ∩ C) = {2} ∪ {3} = {2, 3}
De Morgan’s Laws
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
4. De Morgan’s Law 1
Statement: (A ∪ B)’ = A’ ∩ B’
Proof: Let x ∈ (A ∪ B)’. Then x ∉ A ∪ B ⇒ x ∉ A and x ∉ B ⇒ x ∈ A’ and x ∈ B’ ⇒ x ∈ A’ ∩ B’.
A ∪ B = {1, 2, 3}, (A ∪ B)’ = {4, 5}
A’ = {3, 4, 5}, B’ = {1, 4, 5} ⇒ A’ ∩ B’ = {4, 5}
5. De Morgan’s Law 2
Statement: (A ∩ B)’ = A’ ∪ B’
Proof: Let x ∈ (A ∩ B)’. Then x ∉ A ∩ B ⇒ x ∉ A or x ∉ B ⇒ x ∈ A’ or x ∈ B’ ⇒ x ∈ A’ ∪ B’.
A ∩ B = {2}, (A ∩ B)’ = {1, 3, 4, 5}
A’ = {3, 4, 5}, B’ = {1, 4, 5} ⇒ A’ ∪ B’ = {1, 3, 4, 5}
6. Identity Laws
Statements:
A ∪ ∅ = A
A ∩ U = A
Explanation: Union with an empty set doesn’t add elements; intersection with the universal set keeps all elements of A.
A ∪ ∅ = {1, 2}, A ∩ U = {1, 2}
Example:
If A = {1,2,3}, B = {3,4,5}, U = {1,2,3,4,5,6}
Find A′, B′, A ∪ B, A ∩ B, A − B
Solution:
- A′ = {4,5,6}
- B′ = {1,2,6}
- A ∪ B = {1,2,3,4,5}
- A ∩ B = {3}
- A − B = {1,2}
5. Practical Problems on Union and Intersection (Difficult Level)
Apply the union and intersection formula in challenging, real-life inspired problems.
Formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Example 1:
Out of 150 employees in a company, 80 know Python, 70 know Java, and 40 know both. How many know either Python or Java?
n(A ∪ B) = 80 + 70 − 40 = 110 employees
Example 2:
In a library, 120 people read newspapers, 90 read magazines, and 50 read both. Find how many read either newspapers or magazines.
n(A ∪ B) = 120 + 90 − 50 = 160 readers
Example 3:
In a tech survey, 180 users used iOS, 160 used Android, and 100 used both. How many used at least one of the two platforms?
n(A ∪ B) = 180 + 160 − 100 = 240 users
Example 4:
Out of 300 travelers, 180 visited France, 160 visited Italy, and 100 visited both. How many visited at least one of these countries?
n(A ∪ B) = 180 + 160 − 100 = 240 travelers
Example 5:
Among 250 gamers, 140 play action games, 130 play strategy games, and 80 play both. How many play either action or strategy games?
n(A ∪ B) = 140 + 130 − 80 = 190 gamers
Example 6:
In a survey of 500 shoppers, 300 bought groceries, 250 bought clothes, and 180 bought both. How many bought either groceries or clothes?
n(A ∪ B) = 300 + 250 − 180 = 370 shoppers
Example 7:
Of 200 volunteers, 120 joined environmental causes, 100 joined social causes, and 70 participated in both. How many joined at least one cause?
n(A ∪ B) = 120 + 100 − 70 = 150 volunteers
Example 8:
From a group of 180 authors, 100 write fiction, 90 write non-fiction, and 50 write both. How many write at least one genre?
n(A ∪ B) = 100 + 90 − 50 = 140 authors
Example 9:
In a batch of 400 students, 260 chose Economics, 180 chose Statistics, and 120 chose both. How many chose either subject?
n(A ∪ B) = 260 + 180 − 120 = 320 students
Example 10:
Out of 350 survey participants, 200 use laptops, 170 use tablets, and 90 use both. How many use at least one of the two devices?
n(A ∪ B) = 200 + 170 − 90 = 280 participants
Set Theory Practice Test
Attempt all questions. This test includes questions from all major set operations and properties. Answers are provided at the end for self-evaluation.
Question 1:
In a group of 300 students, 180 play football, 150 play cricket, and 90 play both. How many play either football or cricket?
Question 2:
Out of 600 survey participants, 420 liked online classes. How many did not like online classes?
Question 3:
Out of 250 workers, 180 used protective gear and 120 used both protective gear and helmets. How many used only protective gear?
Question 4:
If Set A = {3, 6, 9} and Set B = {3, 6, 9, 12}, is A a proper subset of B?
Question 5:
Find the number of elements in the power set of A = {1, 2, 3, 4}.
Question 6:
If Set A = {apple, banana, mango} and Set B = {pen, pencil, eraser}, are A and B disjoint?
Question 7:
Set A = {x | x is an even number between 2 and 20}. What is the cardinality of A?
Question 8:
In a survey of 500 people: 300 use WhatsApp, 250 use Instagram, 200 use both, and 100 use neither. How many use either WhatsApp or Instagram?
Answer Key
- Question 1: n(A ∪ B) = 180 + 150 − 90 = 240 students
- Question 2: n(A′) = 600 − 420 = 180 participants
- Question 3: n(A − B) = 180 − 120 = 60 workers
- Question 4: Yes, A ⊂ B (proper subset since A ≠ B)
- Question 5: Power set size = 2⁴ = 16 subsets
- Question 6: Yes, A ∩ B = ∅. So, disjoint sets
- Question 7: Even numbers between 2 and 20: {2, 4, 6, 8, 10, 12, 14, 16, 18} → n(A) = 9
- Question 8: Total using W or I = 500 − 100 = 400 people