Subsets worksheet is much useful to the students who woulds like to practice problems on set theory.

1) Let A = {1, 2, 3, 4, 5} and B = { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

2) Let A = {1, 2, 3, 4, 5} and B = {1, 2, 5}. Determine whether B is a proper subset of A.

3) Let A = {1, 2, 3, 4, 5} find the number of proper subsets of A.

4) Let A = {1, 2, 3 } find the power set of A.

5) Let A = {a, b, c, d, e} find the cardinality of power set of A.

**Problem 1 :**

Let A = {1, 2, 3, 4, 5} and B = { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A. But B is equal A.

**Hence, B is the subset of A, but not a proper subset. **

Let us look at the next problem on "Subsets worksheet"

**Problem 2 :**

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 5}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A.

And also But B is not equal to A.

**Hence, B is a proper subset of A. **

Let us look at the next problem on "Subsets worksheet"

**Problem 3 :**

Let A = {1, 2, 3, 4, 5} find the number of proper subsets of A.

**Solution : **

Let the given set contains "n" number of elements.

Then, the formula to find number of proper subsets is

**= ****2ⁿ****⁻¹**

The value of "n" for the given set A is "5".

Because the set A = {1, 2, 3, 4, 5} contains "5" elements.

Number of proper subsets = 2⁵⁻¹

= 2⁴

= 16

**Hence, the number of proper subsets of A is 16.**

Let us look at the next problem on "Subsets worksheet"

**Problem 4 :**

Let A = {1, 2, 3 } find the power set of A.

**Solution : **

We know that the power set is the set of all subsets.

Here, the given set A contains 3 elements.

Then, the number of subsets = 2³ = 8

Therefore,

**P(A)** = **{**** {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }**** }**

Let us look at the next problem on "Subsets worksheet"

**Problem 5 :**

Let A = {a, b, c, d, e} find the cardinality of power set of A

**Solution : **

The formula for cardinality of power set of A is given below.

**n[P(A)] = 2ⁿ**

Here "n" stands for the number of elements contained by the given set A.

The given set A contains "5" elements. So n = 5.

Then, we have

n[P(A)] = 2⁵

n[P(A)] = 32

**Hence, the cardinality of the power set of A is 32. **

Apart from "Subsets worksheet", let us look at some other worksheets on "Set theory"

1. Which of the following are sets ?

(i) The collection of good books

(ii) The collection of prime numbers less than 30

(iii) The collection of ten most talented mathematics teachers.

(iv) The collection of all students in your school

(v) The collection of all even numbers

2. Let A = {0, 1, 2, 3, 4, 5}. Insert the appropriate symbol ∈ or ∉ in the blank spaces.

(i) 0 ----- A

(ii) 6 ----- A

(iii) 3 ----- A

(iv) 4 ----- A

(v) 7 ----- A

1) Answers

(i) Not a set

(ii) Set

(iii) Not a set

(iv) Set

(v) Set

2) Answers

(i) 0 ∈ A

(ii) 6 ∉ A

(iii) 3 ∈ A

(iv) 4 ∈ A

(v) 7 ∉ A

Write the following sets in Set-Builder form

(i) The set of all positive even numbers

(ii) The set of all whole numbers less than 20

(iii) The set of all positive integers which are multiples of 3

(iv) The set of all odd natural numbers less than 15

(v) The set of all letters in the word ‘computer’

(i) {x : x is a positive even number}

(ii) {x : x is a whole number and x < 20}

(iii) {x : x is a positive integer and multiple of 3}

(iv) {x : x is an odd natural number and x < 15}

(v) {x : x is a letter in the word 'computer'}

Write the following sets in Roster form

1) A = { x : x ∈ N, 2 < x ≤ 10 }

2) B = { x : x ∈ Z, (-1/2) < x < 11/2 }

3) C = {x : x is a prime number and a divisor of 6 }

4) D = { x : x = 2ⁿ, n ∈ N and n ≤ 5 }

5) E = { x : x = 2y - 1, y ≤ 5 y ∈ W }

6) F = { x : x is an integer, x**² **≤ 16 }

1) A = { 3, 4, 5, 6, 7, 8, 9, 10 }

2) B = { 0, 1, 2, 3, 4, 5 }

3) C = { 2, 3 }

4) D = { 2, 4, 8, 16, 32 }

5) E = { -1, 1, 3, 5, 7, 9 }

6) F = { -4, -3, -2, -1, 0, 1, 2, 3, 4 }

Write the following sets in Descriptive form

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

(ii) B = {1, 3, 5, 7, 9, 11}

(iii) C = {1, 4, 9, 16, 25}

(iv) P = {x : x is a letter in the word ‘set theory’}

(v) Q = {x : x is a prime number between 10 and 20}

(i) A is the set of all vowels in the English alphabet

(ii) B is the set of all odd natural numbers less than or equal to 11

(iii) C is the set of all square numbers less than 26.

(iv) P is the set of all letters in the word ‘set theory’

(v) Q is the set of all prime numbers between 10 and 20

1) **In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken **** chemistry and physics course, and 14 had taken all the three courses.** Find how many had taken one course only.

2) **In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games. Find the total number of students in the group. **(Assume that each student in the group plays at least one game.)

3) In a class of 60 students, 40 students like math, 36 like science, 24 like both the subjects. Find the number of students who like

(i) Math only, (ii) Science only (iii) Either Math or Science (iv) Neither Math nor science

**Answer for question (1) :**

**Step 1 :**

Venn diagram related to the information given in the question:

**Step 2 :**

From the venn diagram above, we have

No. of students who had taken only math = 24

No. of students who had taken only chemistry = 60

No. of students who had taken only physics = 22

**Step 3 :**

Total no. of students who had taken only one course

= 24 + 60 + 22

= 106

**Hence, the total number of students who had taken only one course is 106**

**Answer for question (2) :**

**Step 1 :**

Venn diagram related to the information given in the question:

**Step 2 :**

Total number of students in the group

= 28 + 12 + 18 + 7 + 10 + 17 + 8

= 100

**Hence, the total number of students in the group is 100**

**Answer for question (3) :**

**Step 1 :**

Let M and S represent the set of students who like math and science respectively.

**Step 2 :**

From the information given in the question, we have

n(M) = 40, n(S) = 36, n(MnS) = 24

**Step 3 :**

**Answer (i) :** No. of students who like math only

= n(M) - n(MnS)

= 40 - 24

= **16**

**Step 4 :**

**Answer (ii) :** No. of students who like science only

= n(S) - n(MnS)

= 36 - 24

= **12**

**Step 5 :**

**Answer (iii) :** No. of students who like either math or science

= n(M or S)

= n(MuS)

= n(M) + n(S) - n(MnS)

= 40 + 36 - 24

= **52**

**Step 6 :**

**Answer (iv) :**

Total no. students who like any of the two subjects = n(MuS) = 52

No. of students who like neither math nor science

= 60 - 52

= **8**

After having gone through the stuff given above, we hope that the students would have understood "Subsets worksheet".

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