Set Theory Problems - 1 Mark Questions
Set Operations and Representations
1
If A = {x : x ∈ N and x < 20} and B = {x : x ∈ N and x ≤ 5} then write the set A – B in the set builder form.
(M'15)
Step 1: Find the elements of set A
A = {x : x ∈ N and x < 20}
∴ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
A = {x : x ∈ N and x < 20}
∴ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
Step 2: Find the elements of set B
B = {x : x ∈ N and x ≤ 5}
∴ B = {1, 2, 3, 4, 5}
B = {x : x ∈ N and x ≤ 5}
∴ B = {1, 2, 3, 4, 5}
Step 3: Find A - B
A - B = {x : x ∈ A and x ∉ B}
∴ A - B = {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
A - B = {x : x ∈ A and x ∉ B}
∴ A - B = {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
Step 4: Write in set builder form
A - B = {x : x ∈ N, 5 < x < 20}
A - B = {x : x ∈ N, 5 < x < 20}
∴ A - B = {x : x ∈ N, 5 < x < 20}
2
"B is a set of all months in a year having 30 days". Write the above set in the roster form.
(J'15)
Step 1: Identify months with 30 days
Months with 30 days: April, June, September, November
Months with 30 days: April, June, September, November
Step 2: Write in roster form
B = {April, June, September, November}
B = {April, June, September, November}
∴ B = {April, June, September, November}
3
If A – B = {3, 4, 5}, B – A = {1, 8, 9} and A ∩ B = {6, 7}, then find A ∪ B.
(J'15)
Step 1: Understand the given information
A - B = {3, 4, 5} → Elements in A but not in B
B - A = {1, 8, 9} → Elements in B but not in A
A ∩ B = {6, 7} → Elements common to both A and B
A - B = {3, 4, 5} → Elements in A but not in B
B - A = {1, 8, 9} → Elements in B but not in A
A ∩ B = {6, 7} → Elements common to both A and B
Step 2: Find A ∪ B
A ∪ B = (A - B) ∪ (A ∩ B) ∪ (B - A)
A ∪ B = {3, 4, 5} ∪ {6, 7} ∪ {1, 8, 9}
A ∪ B = (A - B) ∪ (A ∩ B) ∪ (B - A)
A ∪ B = {3, 4, 5} ∪ {6, 7} ∪ {1, 8, 9}
Step 3: Combine all elements
A ∪ B = {1, 3, 4, 5, 6, 7, 8, 9}
A ∪ B = {1, 3, 4, 5, 6, 7, 8, 9}
∴ A ∪ B = {1, 3, 4, 5, 6, 7, 8, 9}
4
If A = {1, 1/4, 1/9, 1/16, 1/25}, then write A in set builder form.
(M'16)
Step 1: Observe the pattern
Elements: 1, 1/4, 1/9, 1/16, 1/25
These can be written as: 1/1², 1/2², 1/3², 1/4², 1/5²
Elements: 1, 1/4, 1/9, 1/16, 1/25
These can be written as: 1/1², 1/2², 1/3², 1/4², 1/5²
Step 2: Write in set builder form
A = {1/n² : n ∈ N, 1 ≤ n ≤ 5}
A = {1/n² : n ∈ N, 1 ≤ n ≤ 5}
∴ A = {1/n² : n ∈ N, 1 ≤ n ≤ 5}
5
A = {x: x ∈ N, x is a composite number and x < 13}. Write set A in the roster form.
(J'16)
Step 1: Identify composite numbers less than 13
Composite numbers are positive integers that have at least one divisor other than 1 and themselves.
Composite numbers are positive integers that have at least one divisor other than 1 and themselves.
Step 2: List composite numbers < 13
4, 6, 8, 9, 10, 12
4, 6, 8, 9, 10, 12
Step 3: Write in roster form
A = {4, 6, 8, 9, 10, 12}
A = {4, 6, 8, 9, 10, 12}
∴ A = {4, 6, 8, 9, 10, 12}
6
Represent A ∩ B through Venn diagram, where A = {1, 4, 6, 9, 10} and B = {x / x is a perfect square less than 25}.
(M'17)
Step 1: Find elements of set B
B = {x / x is a perfect square less than 25}
Perfect squares less than 25: 1, 4, 9, 16
∴ B = {1, 4, 9, 16}
B = {x / x is a perfect square less than 25}
Perfect squares less than 25: 1, 4, 9, 16
∴ B = {1, 4, 9, 16}
Step 2: Find A ∩ B
A = {1, 4, 6, 9, 10}
B = {1, 4, 9, 16}
A ∩ B = {1, 4, 9}
A = {1, 4, 6, 9, 10}
B = {1, 4, 9, 16}
A ∩ B = {1, 4, 9}
Step 3: Describe the Venn diagram
In a Venn diagram, A ∩ B would be represented by the overlapping region of two circles, one for set A and one for set B, containing the elements {1, 4, 9}.
In a Venn diagram, A ∩ B would be represented by the overlapping region of two circles, one for set A and one for set B, containing the elements {1, 4, 9}.
∴ A ∩ B = {1, 4, 9}
7
If A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6}, Find A ∩ B.
(J'17)
Step 1: Identify common elements
A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6}
A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6}
Step 2: Find intersection
A ∩ B = {x : x ∈ A and x ∈ B}
A ∩ B = {3, 4, 5}
A ∩ B = {x : x ∈ A and x ∈ B}
A ∩ B = {3, 4, 5}
∴ A ∩ B = {3, 4, 5}
8
Give one example each for a finite set and an infinite set.
(M'18)
Finite Set Example:
A set with a countable number of elements.
Example: A = {1, 2, 3, 4, 5} or B = {x : x is a day of the week}
A set with a countable number of elements.
Example: A = {1, 2, 3, 4, 5} or B = {x : x is a day of the week}
Infinite Set Example:
A set with an unlimited number of elements.
Example: N = {1, 2, 3, 4, ...} or Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
A set with an unlimited number of elements.
Example: N = {1, 2, 3, 4, ...} or Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
∴ Finite Set: {1, 2, 3, 4, 5}
Infinite Set: {1, 2, 3, 4, ...}
Infinite Set: {1, 2, 3, 4, ...}
9
List all the subsets of the set A = {x, y, z}
(J'18)
Step 1: Subsets with 0 elements (Empty set)
∅ or {}
∅ or {}
Step 2: Subsets with 1 element
{x}, {y}, {z}
{x}, {y}, {z}
Step 3: Subsets with 2 elements
{x, y}, {x, z}, {y, z}
{x, y}, {x, z}, {y, z}
Step 4: Subset with 3 elements (The set itself)
{x, y, z}
{x, y, z}
∴ All subsets: ∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}
10
If A = {x: x is a factor of 24}, then find n(A).
(M'19)
Step 1: Find factors of 24
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Step 2: Write set A
A = {1, 2, 3, 4, 6, 8, 12, 24}
A = {1, 2, 3, 4, 6, 8, 12, 24}
Step 3: Find n(A) - number of elements in A
n(A) = 8
n(A) = 8
∴ n(A) = 8
11
If A = {1, 2, 3}, B = {3, 4, 5} Then find A - B and B - A.
(J'19)
Step 1: Find A - B
A - B = {x : x ∈ A and x ∉ B}
A = {1, 2, 3}, B = {3, 4, 5}
A - B = {1, 2}
A - B = {x : x ∈ A and x ∉ B}
A = {1, 2, 3}, B = {3, 4, 5}
A - B = {1, 2}
Step 2: Find B - A
B - A = {x : x ∈ B and x ∉ A}
B - A = {4, 5}
B - A = {x : x ∈ B and x ∉ A}
B - A = {4, 5}
∴ A - B = {1, 2}
B - A = {4, 5}
B - A = {4, 5}
12
A = {x : x is a factor of 8}, B = {x : x is a factor of 36}. Is A ⊂ B? Justify.
(Jun'23)
Step 1: Find elements of A
Factors of 8: 1, 2, 4, 8
∴ A = {1, 2, 4, 8}
Factors of 8: 1, 2, 4, 8
∴ A = {1, 2, 4, 8}
Step 2: Find elements of B
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
∴ B = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
∴ B = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Step 3: Check if A ⊂ B
A ⊂ B if every element of A is also an element of B.
Check each element of A:
1 ∈ B, 2 ∈ B, 4 ∈ B, 8 ∈ B?
8 is NOT an element of B
A ⊂ B if every element of A is also an element of B.
Check each element of A:
1 ∈ B, 2 ∈ B, 4 ∈ B, 8 ∈ B?
8 is NOT an element of B
∴ A is NOT a subset of B because 8 ∈ A but 8 ∉ B
Set Theory Problems - 2 Mark Questions
1
If A = {x : x ∈ N and x < 6} and B = {x : x ∈ N and 3 < x < 8} then Show that A – B ≠ B – A with the help of Venn diagram.
(M'15)
Step 1: Find the elements of set A
A = {x : x ∈ N and x < 6}
∴ A = {1, 2, 3, 4, 5}
A = {x : x ∈ N and x < 6}
∴ A = {1, 2, 3, 4, 5}
Step 2: Find the elements of set B
B = {x : x ∈ N and 3 < x < 8}
∴ B = {4, 5, 6, 7}
B = {x : x ∈ N and 3 < x < 8}
∴ B = {4, 5, 6, 7}
Step 3: Find A - B
A - B = {x : x ∈ A and x ∉ B}
A - B = {1, 2, 3}
A - B = {x : x ∈ A and x ∉ B}
A - B = {1, 2, 3}
Step 4: Find B - A
B - A = {x : x ∈ B and x ∉ A}
B - A = {6, 7}
B - A = {x : x ∈ B and x ∉ A}
B - A = {6, 7}
Step 5: Venn Diagram Description
In the Venn diagram:
- Circle A contains elements: {1, 2, 3, 4, 5}
- Circle B contains elements: {4, 5, 6, 7}
- The overlapping region (A ∩ B) contains: {4, 5}
- A - B is represented by the region in A but not in B: {1, 2, 3}
- B - A is represented by the region in B but not in A: {6, 7}
These two regions are clearly different.
- Circle A contains elements: {1, 2, 3, 4, 5}
- Circle B contains elements: {4, 5, 6, 7}
- The overlapping region (A ∩ B) contains: {4, 5}
- A - B is represented by the region in A but not in B: {1, 2, 3}
- B - A is represented by the region in B but not in A: {6, 7}
These two regions are clearly different.
∴ A - B = {1, 2, 3} and B - A = {6, 7}
Since {1, 2, 3} ≠ {6, 7}, we have shown that A - B ≠ B - A.
Since {1, 2, 3} ≠ {6, 7}, we have shown that A - B ≠ B - A.
2
Answer the following questions and justify your answers.
A = {x : x ∈ N, x < 2015}, is it a finite set or infinite set?
B = {x : x + 5 = 5} is it a null set or a Universal set?
(J'15)
For Set A:
A = {x : x ∈ N, x < 2015}
This means A contains all natural numbers less than 2015.
These are: 1, 2, 3, ..., 2014
The count is finite (2014 elements).
A = {x : x ∈ N, x < 2015}
This means A contains all natural numbers less than 2015.
These are: 1, 2, 3, ..., 2014
The count is finite (2014 elements).
For Set B:
B = {x : x + 5 = 5}
Solving x + 5 = 5, we get x = 0
So B = {0}
This is not an empty set as it contains one element (0).
It's also not a universal set as it doesn't contain all possible elements.
B = {x : x + 5 = 5}
Solving x + 5 = 5, we get x = 0
So B = {0}
This is not an empty set as it contains one element (0).
It's also not a universal set as it doesn't contain all possible elements.
∴ A is a finite set because it has a finite number of elements (2014).
B is neither a null set nor a universal set - it's a singleton set containing {0}.
B is neither a null set nor a universal set - it's a singleton set containing {0}.
3
A = {x : x ∈ N, and x is a factor of 30};
B = {x : x ∈ N, and x is a prime factor of 30}
draw Venn diagram for A∪B
(J'16)
Step 1: Find elements of set A
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
∴ A = {1, 2, 3, 5, 6, 10, 15, 30}
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
∴ A = {1, 2, 3, 5, 6, 10, 15, 30}
Step 2: Find elements of set B
Prime factors of 30: 2, 3, 5
∴ B = {2, 3, 5}
Prime factors of 30: 2, 3, 5
∴ B = {2, 3, 5}
Step 3: Find A ∪ B
A ∪ B = {1, 2, 3, 5, 6, 10, 15, 30}
A ∪ B = {1, 2, 3, 5, 6, 10, 15, 30}
Step 4: Venn Diagram Description
In the Venn diagram:
- Circle A contains: {1, 2, 3, 5, 6, 10, 15, 30}
- Circle B contains: {2, 3, 5}
- Since all elements of B are already in A, circle B is completely inside circle A
- The union A ∪ B is represented by the entire circle A
- The overlapping region (A ∩ B) is exactly equal to B: {2, 3, 5}
- Circle A contains: {1, 2, 3, 5, 6, 10, 15, 30}
- Circle B contains: {2, 3, 5}
- Since all elements of B are already in A, circle B is completely inside circle A
- The union A ∪ B is represented by the entire circle A
- The overlapping region (A ∩ B) is exactly equal to B: {2, 3, 5}
∴ A ∪ B = {1, 2, 3, 5, 6, 10, 15, 30}
4
If A = {x : x ∈ N, x < 10}, B = {x : x is a prime number and x < 10},
Then show that A – B ≠ B – A with the help of Venn diagram.
(J'17)
Step 1: Find the elements of set A
A = {x : x ∈ N, x < 10}
∴ A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {x : x ∈ N, x < 10}
∴ A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Step 2: Find the elements of set B
B = {x : x is a prime number and x < 10}
Prime numbers less than 10: 2, 3, 5, 7
∴ B = {2, 3, 5, 7}
B = {x : x is a prime number and x < 10}
Prime numbers less than 10: 2, 3, 5, 7
∴ B = {2, 3, 5, 7}
Step 3: Find A - B
A - B = {x : x ∈ A and x ∉ B}
A - B = {1, 4, 6, 8, 9}
A - B = {x : x ∈ A and x ∉ B}
A - B = {1, 4, 6, 8, 9}
Step 4: Find B - A
B - A = {x : x ∈ B and x ∉ A}
Since all elements of B are in A, B - A = ∅
B - A = {x : x ∈ B and x ∉ A}
Since all elements of B are in A, B - A = ∅
Step 5: Venn Diagram Description
In the Venn diagram:
- Circle A contains: {1, 2, 3, 4, 5, 6, 7, 8, 9}
- Circle B contains: {2, 3, 5, 7}
- The overlapping region (A ∩ B) contains: {2, 3, 5, 7}
- A - B is represented by the region in A but not in B: {1, 4, 6, 8, 9}
- B - A is represented by the region in B but not in A: ∅ (empty)
These two regions are clearly different.
- Circle A contains: {1, 2, 3, 4, 5, 6, 7, 8, 9}
- Circle B contains: {2, 3, 5, 7}
- The overlapping region (A ∩ B) contains: {2, 3, 5, 7}
- A - B is represented by the region in A but not in B: {1, 4, 6, 8, 9}
- B - A is represented by the region in B but not in A: ∅ (empty)
These two regions are clearly different.
∴ A - B = {1, 4, 6, 8, 9} and B - A = ∅
Since {1, 4, 6, 8, 9} ≠ ∅, we have shown that A - B ≠ B - A.
Since {1, 4, 6, 8, 9} ≠ ∅, we have shown that A - B ≠ B - A.
5
If A = {1, 2, 3, 4}, B = {2, 4, 6, 8, 10}, then represent the Venn diagram of A - B.
(J'18)
Step 1: Find A - B
A - B = {x : x ∈ A and x ∉ B}
A = {1, 2, 3, 4}, B = {2, 4, 6, 8, 10}
A - B = {1, 3}
A - B = {x : x ∈ A and x ∉ B}
A = {1, 2, 3, 4}, B = {2, 4, 6, 8, 10}
A - B = {1, 3}
Step 2: Venn Diagram Description
In the Venn diagram:
- Circle A contains: {1, 2, 3, 4}
- Circle B contains: {2, 4, 6, 8, 10}
- The overlapping region (A ∩ B) contains: {2, 4}
- A - B is represented by the region in A but not in B: {1, 3}
This region should be shaded or highlighted to represent A - B.
- Circle A contains: {1, 2, 3, 4}
- Circle B contains: {2, 4, 6, 8, 10}
- The overlapping region (A ∩ B) contains: {2, 4}
- A - B is represented by the region in A but not in B: {1, 3}
This region should be shaded or highlighted to represent A - B.
∴ A - B = {1, 3}
6
If μ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 3, 5, 8} and B = {0, 3, 5, 7, 10}.
Then represent A∩B in the Venn diagram.
(M'19)
Step 1: Find A ∩ B
A ∩ B = {x : x ∈ A and x ∈ B}
A = {2, 3, 5, 8}, B = {0, 3, 5, 7, 10}
A ∩ B = {3, 5}
A ∩ B = {x : x ∈ A and x ∈ B}
A = {2, 3, 5, 8}, B = {0, 3, 5, 7, 10}
A ∩ B = {3, 5}
Step 2: Venn Diagram Description
In the Venn diagram with universal set μ:
- There is a rectangle representing the universal set μ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- Circle A contains: {2, 3, 5, 8}
- Circle B contains: {0, 3, 5, 7, 10}
- The overlapping region (A ∩ B) contains: {3, 5}
This overlapping region should be shaded or highlighted to represent A ∩ B.
- There is a rectangle representing the universal set μ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- Circle A contains: {2, 3, 5, 8}
- Circle B contains: {0, 3, 5, 7, 10}
- The overlapping region (A ∩ B) contains: {3, 5}
This overlapping region should be shaded or highlighted to represent A ∩ B.
∴ A ∩ B = {3, 5}
7
If A = {x : x is a factor of 12} and B = {x: x is a factor of 6} then find A∪B and A∩B.
(J'19)
Step 1: Find elements of set A
Factors of 12: 1, 2, 3, 4, 6, 12
∴ A = {1, 2, 3, 4, 6, 12}
Factors of 12: 1, 2, 3, 4, 6, 12
∴ A = {1, 2, 3, 4, 6, 12}
Step 2: Find elements of set B
Factors of 6: 1, 2, 3, 6
∴ B = {1, 2, 3, 6}
Factors of 6: 1, 2, 3, 6
∴ B = {1, 2, 3, 6}
Step 3: Find A ∪ B
A ∪ B = {1, 2, 3, 4, 6, 12}
A ∪ B = {1, 2, 3, 4, 6, 12}
Step 4: Find A ∩ B
A ∩ B = {1, 2, 3, 6}
A ∩ B = {1, 2, 3, 6}
∴ A ∪ B = {1, 2, 3, 4, 6, 12}
A ∩ B = {1, 2, 3, 6}
A ∩ B = {1, 2, 3, 6}
8
If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}.
Then show that n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
(May 2022)
Step 1: Find n(A)
A = {1, 2, 3, 4, 5}
n(A) = 5
A = {1, 2, 3, 4, 5}
n(A) = 5
Step 2: Find n(B)
B = {2, 4, 6, 8}
n(B) = 4
B = {2, 4, 6, 8}
n(B) = 4
Step 3: Find A ∩ B
A ∩ B = {2, 4}
n(A ∩ B) = 2
A ∩ B = {2, 4}
n(A ∩ B) = 2
Step 4: Find A ∪ B
A ∪ B = {1, 2, 3, 4, 5, 6, 8}
n(A ∪ B) = 7
A ∪ B = {1, 2, 3, 4, 5, 6, 8}
n(A ∪ B) = 7
Step 5: Verify the formula
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
7 = 5 + 4 - 2
7 = 7
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
7 = 5 + 4 - 2
7 = 7
∴ The formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B) is verified
as 7 = 5 + 4 - 2.
as 7 = 5 + 4 - 2.
9
If A = {4, 8, 12, 16, 20}, B = {6, 12, 18, 24, 30},
then show that n(A∪B) = n(A) + n(B) – n(A∩B).
(Aug 2022)
Step 1: Find n(A)
A = {4, 8, 12, 16, 20}
n(A) = 5
A = {4, 8, 12, 16, 20}
n(A) = 5
Step 2: Find n(B)
B = {6, 12, 18, 24, 30}
n(B) = 5
B = {6, 12, 18, 24, 30}
n(B) = 5
Step 3: Find A ∩ B
A ∩ B = {12}
n(A ∩ B) = 1
A ∩ B = {12}
n(A ∩ B) = 1
Step 4: Find A ∪ B
A ∪ B = {4, 6, 8, 12, 16, 18, 20, 24, 30}
n(A ∪ B) = 9
A ∪ B = {4, 6, 8, 12, 16, 18, 20, 24, 30}
n(A ∪ B) = 9
Step 5: Verify the formula
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
9 = 5 + 5 - 1
9 = 9
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
9 = 5 + 5 - 1
9 = 9
∴ The formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B) is verified
as 9 = 5 + 5 - 1.
as 9 = 5 + 5 - 1.
Set Theory Problems and Solutions
4-Mark Questions
1
X is a set of factors of 24 and Y is a set of factors of 36, then find sets X∪Y and X∩Y by using Venn diagram and comment on the answer.
(M'16)
Step 1: Find elements of set X
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
∴ X = {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
∴ X = {1, 2, 3, 4, 6, 8, 12, 24}
Step 2: Find elements of set Y
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
∴ Y = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
∴ Y = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Step 3: Find X ∪ Y
X ∪ Y = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}
X ∪ Y = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}
Step 4: Find X ∩ Y
X ∩ Y = {1, 2, 3, 4, 6, 12}
X ∩ Y = {1, 2, 3, 4, 6, 12}
Step 5: Venn Diagram Description
In the Venn diagram:
- Circle X contains: {1, 2, 3, 4, 6, 8, 12, 24}
- Circle Y contains: {1, 2, 3, 4, 6, 9, 12, 18, 36}
- The overlapping region (X ∩ Y) contains: {1, 2, 3, 4, 6, 12}
- Elements only in X: {8, 24}
- Elements only in Y: {9, 18, 36}
- Circle X contains: {1, 2, 3, 4, 6, 8, 12, 24}
- Circle Y contains: {1, 2, 3, 4, 6, 9, 12, 18, 36}
- The overlapping region (X ∩ Y) contains: {1, 2, 3, 4, 6, 12}
- Elements only in X: {8, 24}
- Elements only in Y: {9, 18, 36}
∴ X ∪ Y = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}
X ∩ Y = {1, 2, 3, 4, 6, 12}
Comment: X ∩ Y represents the common factors of 24 and 36, which are the factors of their GCD (12).
X ∩ Y = {1, 2, 3, 4, 6, 12}
Comment: X ∩ Y represents the common factors of 24 and 36, which are the factors of their GCD (12).
2
A = {x : x ∈ N and x is a multiple of 4}; B = {x : x ∈ N and x is a multiple of 6};
C = {x : x ∈ N and x is a multiple LCM of 4 and 6}. Find A∩B. How can you relate the sets A∩B and C.
(J'16)
Step 1: Find A ∩ B
A = {4, 8, 12, 16, 20, 24, 28, 32, 36, ...}
B = {6, 12, 18, 24, 30, 36, 42, ...}
A ∩ B = {12, 24, 36, 48, ...}
A = {4, 8, 12, 16, 20, 24, 28, 32, 36, ...}
B = {6, 12, 18, 24, 30, 36, 42, ...}
A ∩ B = {12, 24, 36, 48, ...}
Step 2: Find set C
LCM of 4 and 6 is 12
C = {x : x ∈ N and x is a multiple of 12}
C = {12, 24, 36, 48, 60, ...}
LCM of 4 and 6 is 12
C = {x : x ∈ N and x is a multiple of 12}
C = {12, 24, 36, 48, 60, ...}
Step 3: Compare A ∩ B and C
A ∩ B = {12, 24, 36, 48, ...}
C = {12, 24, 36, 48, ...}
A ∩ B = {12, 24, 36, 48, ...}
C = {12, 24, 36, 48, ...}
∴ A ∩ B = {12, 24, 36, 48, ...}
C = {12, 24, 36, 48, ...}
Therefore, A ∩ B = C
The intersection of multiples of 4 and multiples of 6 gives us multiples of their LCM (12).
C = {12, 24, 36, 48, ...}
Therefore, A ∩ B = C
The intersection of multiples of 4 and multiples of 6 gives us multiples of their LCM (12).
3
A = {x : x is a perfect square, x < 50, x ∈ N}, B = {x : x = 8m + 1, where m ∈ W, x < 50, x ∈ N}.
Find A∩B and display it with Venn diagram.
(M'18)
Step 1: Find elements of set A
Perfect squares less than 50: 1, 4, 9, 16, 25, 36, 49
∴ A = {1, 4, 9, 16, 25, 36, 49}
Perfect squares less than 50: 1, 4, 9, 16, 25, 36, 49
∴ A = {1, 4, 9, 16, 25, 36, 49}
Step 2: Find elements of set B
B = {x : x = 8m + 1, where m ∈ W, x < 50}
For m = 0: 8×0 + 1 = 1
For m = 1: 8×1 + 1 = 9
For m = 2: 8×2 + 1 = 17
For m = 3: 8×3 + 1 = 25
For m = 4: 8×4 + 1 = 33
For m = 5: 8×5 + 1 = 41
For m = 6: 8×6 + 1 = 49
∴ B = {1, 9, 17, 25, 33, 41, 49}
B = {x : x = 8m + 1, where m ∈ W, x < 50}
For m = 0: 8×0 + 1 = 1
For m = 1: 8×1 + 1 = 9
For m = 2: 8×2 + 1 = 17
For m = 3: 8×3 + 1 = 25
For m = 4: 8×4 + 1 = 33
For m = 5: 8×5 + 1 = 41
For m = 6: 8×6 + 1 = 49
∴ B = {1, 9, 17, 25, 33, 41, 49}
Step 3: Find A ∩ B
A ∩ B = {1, 9, 25, 49}
A ∩ B = {1, 9, 25, 49}
Step 4: Venn Diagram Description
In the Venn diagram:
- Circle A contains: {1, 4, 9, 16, 25, 36, 49}
- Circle B contains: {1, 9, 17, 25, 33, 41, 49}
- The overlapping region (A ∩ B) contains: {1, 9, 25, 49}
- Elements only in A: {4, 16, 36}
- Elements only in B: {17, 33, 41}
- Circle A contains: {1, 4, 9, 16, 25, 36, 49}
- Circle B contains: {1, 9, 17, 25, 33, 41, 49}
- The overlapping region (A ∩ B) contains: {1, 9, 25, 49}
- Elements only in A: {4, 16, 36}
- Elements only in B: {17, 33, 41}
∴ A ∩ B = {1, 9, 25, 49}
4
If A = {x : x is a prime and x < 10}, B = {x : x is a factor of 6}, then find A∩B, A∪B and A – B.
(J'18)
Step 1: Find elements of set A
Prime numbers less than 10: 2, 3, 5, 7
∴ A = {2, 3, 5, 7}
Prime numbers less than 10: 2, 3, 5, 7
∴ A = {2, 3, 5, 7}
Step 2: Find elements of set B
Factors of 6: 1, 2, 3, 6
∴ B = {1, 2, 3, 6}
Factors of 6: 1, 2, 3, 6
∴ B = {1, 2, 3, 6}
Step 3: Find A ∩ B
A ∩ B = {2, 3}
A ∩ B = {2, 3}
Step 4: Find A ∪ B
A ∪ B = {1, 2, 3, 5, 6, 7}
A ∪ B = {1, 2, 3, 5, 6, 7}
Step 5: Find A - B
A - B = {5, 7}
A - B = {5, 7}
∴ A ∩ B = {2, 3}
A ∪ B = {1, 2, 3, 5, 6, 7}
A - B = {5, 7}
A ∪ B = {1, 2, 3, 5, 6, 7}
A - B = {5, 7}
5
If A = {x : 2x + 1, x ∈ N, x ≤ 5}, B = {x : x is a composite number, x ≤ 12},
then show that (A⋃B) – (A⋂B) = (A – B) ⋃ (B – A)
(M'19)
Step 1: Find elements of set A
A = {2x + 1, x ∈ N, x ≤ 5}
For x = 1: 2×1 + 1 = 3
For x = 2: 2×2 + 1 = 5
For x = 3: 2×3 + 1 = 7
For x = 4: 2×4 + 1 = 9
For x = 5: 2×5 + 1 = 11
∴ A = {3, 5, 7, 9, 11}
A = {2x + 1, x ∈ N, x ≤ 5}
For x = 1: 2×1 + 1 = 3
For x = 2: 2×2 + 1 = 5
For x = 3: 2×3 + 1 = 7
For x = 4: 2×4 + 1 = 9
For x = 5: 2×5 + 1 = 11
∴ A = {3, 5, 7, 9, 11}
Step 2: Find elements of set B
Composite numbers ≤ 12: 4, 6, 8, 9, 10, 12
∴ B = {4, 6, 8, 9, 10, 12}
Composite numbers ≤ 12: 4, 6, 8, 9, 10, 12
∴ B = {4, 6, 8, 9, 10, 12}
Step 3: Find A ∪ B
A ∪ B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A ∪ B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Step 4: Find A ∩ B
A ∩ B = {9}
A ∩ B = {9}
Step 5: Find (A ∪ B) - (A ∩ B)
(A ∪ B) - (A ∩ B) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
(A ∪ B) - (A ∩ B) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
Step 6: Find A - B
A - B = {3, 5, 7, 11}
A - B = {3, 5, 7, 11}
Step 7: Find B - A
B - A = {4, 6, 8, 10, 12}
B - A = {4, 6, 8, 10, 12}
Step 8: Find (A - B) ∪ (B - A)
(A - B) ∪ (B - A) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
(A - B) ∪ (B - A) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
∴ (A ∪ B) - (A ∩ B) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
(A - B) ∪ (B - A) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
Since both sets are equal, we have shown that (A∪B) – (A∩B) = (A – B) ∪ (B – A).
(A - B) ∪ (B - A) = {3, 4, 5, 6, 7, 8, 10, 11, 12}
Since both sets are equal, we have shown that (A∪B) – (A∩B) = (A – B) ∪ (B – A).
6
If A = {x : x is a prime less than 20} and B = {x : x is whole number less than 10}
then verify n(A∪B) = n(A) + n(B) – n(A∩B).
(J'19)
Step 1: Find elements of set A
Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19
∴ A = {2, 3, 5, 7, 11, 13, 17, 19}
n(A) = 8
Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19
∴ A = {2, 3, 5, 7, 11, 13, 17, 19}
n(A) = 8
Step 2: Find elements of set B
Whole numbers less than 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
∴ B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
n(B) = 10
Whole numbers less than 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
∴ B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
n(B) = 10
Step 3: Find A ∩ B
A ∩ B = {2, 3, 5, 7}
n(A ∩ B) = 4
A ∩ B = {2, 3, 5, 7}
n(A ∩ B) = 4
Step 4: Find A ∪ B
A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 19}
n(A ∪ B) = 14
A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 19}
n(A ∪ B) = 14
Step 5: Verify the formula
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
14 = 8 + 10 - 4
14 = 14
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
14 = 8 + 10 - 4
14 = 14
∴ The formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B) is verified
as 14 = 8 + 10 - 4.
as 14 = 8 + 10 - 4.
Multiple Choice Questions
1
Set A = {F, L, W, O}. Which of the following is not a set builder form for set A?
(J'15)
Given: A = {F, L, W, O}
Explanation:
- Option A: FOLLOW → {F, O, L, W} = A
- Option B: FLOW → {F, L, O, W} = A
- Option C: WOLF → {W, O, L, F} = A
- Option D: SLOW → {S, L, O, W} ≠ A (contains S instead of F)
Therefore, option D is not a set builder form for set A.
- Option A: FOLLOW → {F, O, L, W} = A
- Option B: FLOW → {F, L, O, W} = A
- Option C: WOLF → {W, O, L, F} = A
- Option D: SLOW → {S, L, O, W} ≠ A (contains S instead of F)
Therefore, option D is not a set builder form for set A.
∴ Correct answer: D
2
If the union of two sets is one of the set itself, then the relation between the two sets is
(J'15)
Given: A ∪ B = A (or A ∪ B = B)
Explanation:
If A ∪ B = A, then all elements of B are already in A, which means B ⊆ A.
Similarly, if A ∪ B = B, then A ⊆ B.
Therefore, one set is a subset of the other.
If A ∪ B = A, then all elements of B are already in A, which means B ⊆ A.
Similarly, if A ∪ B = B, then A ⊆ B.
Therefore, one set is a subset of the other.
∴ Correct answer: A
3
Which one of the following is the example of finite set?
(M'16)
Explanation:
- Option A: {x / x ∈ N and x² = 9} = {3} → Finite set
- Option B: Multiples of even prime numbers (2) = {2, 4, 6, 8, ...} → Infinite set
- Option C: Rational numbers between 2 and 3 = Infinite set
- Option D: Odd prime numbers = {3, 5, 7, 11, 13, ...} → Infinite set
- Option A: {x / x ∈ N and x² = 9} = {3} → Finite set
- Option B: Multiples of even prime numbers (2) = {2, 4, 6, 8, ...} → Infinite set
- Option C: Rational numbers between 2 and 3 = Infinite set
- Option D: Odd prime numbers = {3, 5, 7, 11, 13, ...} → Infinite set
∴ Correct answer: A
