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Edited By Vishal kumar | Updated on Nov 01, 2023 11:46 AM IST

In the previous exercises, you have learned about the set, subset, power set, universal set, Venn's diagram, and operations on the sets. In the NCERT solutions for Class 11 Maths chapter 1 exercise 1.5, you will learn about the complement of a set. The properties of the complement of set like complements law, De Morgan's law, the law of double complementation, laws of empty set and universal set are also covered in the NCERT book exercise 1.5 Class 11 Maths.

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- NCERT Solutions for Class 11 Maths Chapter 1: Sets Exercise 1.5- Download Free PDF
- NCERT Solutions for Class 11 Maths Chapter 1 – Sets Exercise 1.5
- Access Sets Class 11 Chapter 1 Exercise: 1.5
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- NCERT Solutions of Class 11 Subject Wise
- Subject Wise NCERT Exampler Solutions

De Morgan's law states that " the complement of the union of two sets A and B is the intersection of the complement of A and complement of B, and the complement of the intersection of two sets A and B is the union of complements of A and complement of B". You can prove these laws using Venn's diagram that you have learned in the last exercise. There are many questions related to these properties of the complement of sets in the Class 11 Maths chapter 1 exercise 1.5 that you can solve by yourself. This class 11 maths ex 1.5 is a bit difficult as compared to the previous exercise, so you may not be able to solve all the problems at first. You can go through the Class 11 Maths chapter 1 exercise 1.5 solutions to get conceptual clarity. You can check NCERT Solutions if you are looking for NCERT solutions at one place.

**Also, see**

- Sets Exercise 1.1
- Sets Exercise 1.2
- Sets Exercise 1.3
- Sets Exercise 1.4
- Sets Exercise 1.6
- Sets Miscellaneous Exercise

(i) A′

(ii) B′

(iii) (A C)′

(iv) (A B)′

(v) (A')'

(vi) (B – C)'

Answer:

U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }

(i) A′ = U - A = {5,6,7,8,9}

(ii) B′ = U - B = {1,3,5,7,9}

(iii) A C = {1,2,3,4,5,6}

(A C)′ = U - (A C) = {7,8,9}

(iv) (A B) = {1,2,3,4,6,8}

(A B)′ = U - (A B) = {5,7,9}

(v) (A')' = A = { 1, 2, 3, 4}

(vi) (B – C) = {2,8}

(B – C)' = U - (B – C) = {1,3,4,5,6,7,9}

Question:2 If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) A = {a, b, c}

(ii) B = {d, e, f, g}

(iii) C = {a, c, e, g}

(iv) D = { f, g, h, a}

Answer:

U = { a, b, c, d, e, f, g, h}

(i) A = {a, b, c}

A' = U - A = {d,e,f,g,h}

(ii) B = {d, e, f, g}

B' = U - B = {a,b,c,h}

(iii) C = {a, c, e, g}

C' = U - C = {b,d,f,h}

(iv) D = { f, g, h, a}

D' = U - D = {b,c,d,e}

(i) {x : x is an even natural number}

(ii) { x : x is an odd natural number }

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

(iv) { x : x is a prime number }

(v) {x : x is a natural number divisible by 3 and 5}

Answer:

Universal set = U = {1,2,3,4,5,6,7....................}

(i) {x : x is an even natural number} = {2,4,6,8,..........}

{x : x is an even natural number}'= U - {x : x is an even natural number} = {1,3,5,7,9,..........} = {x : x is an odd natural number}

(ii) { x : x is an odd natural number }' = U - { x : x is an odd natural number } = {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}' = U - {x : x is a positive multiple of 3} = {x : x , x N and is not a positive multiple of 3}

(iv) { x : x is a prime number }' = U - { x : x is a prime number } = { x : x is a positive composite number and 1 }

(v) {x : x is a natural number divisible by 3 and 5}' = U - {x : x is a natural number divisible by 3 and 5} = {x : x is a natural number not divisible by 3 or 5}

(vi) { x : x is a perfect square }

(vii) { x : x is a perfect cube}

(viii) { x : x + 5 8 }

(ix) { x : 2x + 5 9}

(x) { x : x 7 }

(xi) { x : x N and 2x + 1 10 }

Answer:

Universal set = U = {1,2,3,4,5,6,7,8.............}

(vi) { x : x is a perfect square }' = U - { x : x is a perfect square } = { x : x N and x is not a perfect square }

(vii) { x : x is a perfect cube}' = U - { x : x is a perfect cube} = { x : x N and x is not a perfect cube}

(viii) { x : x + 5 8 }' = U - { x : x + 5 8 } = U - {3} = { x : x N and x 3 }

(ix) { x : 2x + 5 9}' = U - { x : 2x + 5 9} = U -{2} = { x : x N and x 2}

(x) { x : x 7 }' = U - { x : x 7 } = { x : x N and x 7 }

(xi) { x : x N and 2x + 1 10 }' = U - { x : x N and x 9/2 } = { x : x N and x 9/2 }

Question:4 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that

(i) (A B)′ = A′ B′

(ii) (A B)′ = A′ B′

Answer:

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

(i) (A B)′ = A′ B′

L.H.S = (A B)′ = U - (A B) = {1,9}

R.H.S = A′ B′ = {1,3,5,7,9} {1,4,6,8,9} = {1,9}

L.H.S = R.H.S

Hence,the statement is true.

(ii) (A B)′ = A′ B′

L.H.S = U - (A B) ={1,3,4,5,6,7,8,9}

R.H.S = A′ B′ = {1,3,5,7,9} {1,4,6,8,9} = {1,3,4,5,6,7,8,9}

L.H.S = R.H.S

Hence,the statement is true.

Question:5 Draw appropriate Venn diagram for each of the following :

(i) (A ∪ B)′

(ii) A′ ∩ B′

(iii) (A ∩ B)′

(iv) A′ ∪ B′

Answer:

(i) (A ∪ B)′

(A ∪ B) is in yellow colour

(A ∪ B)′ is in green colour

ii) A′ ∩ B′ is represented by the green colour in the below figure

iii) (A ∩ B)′ is represented by green colour in the below diagram and white colour represents (A ∩ B)

iv) A′ ∪ B′

The green colour represents A′ ∪ B′

Answer:

A' is the set of all triangles whose angle is in other words A' is set of all equilateral triangles.

Question:7 Fill in the blanks to make each of the following a true statement :

(i) A A′

(ii) A

(iii) A A′

(iv) U′ A

Answer:

The following are the answers for the questions

(i) A A′ U

(ii) ′ A A

(iii) A A′

(iv) U′ A

NCERT syllabus Class 11th Maths chapter 1 exercise 1.5 consists of questions related to finding the compliments of sets using the properties of complements of sets. There are some examples and definitions given before this ex 1.5 class 11. You must go through these examples and properties to get more clarity. A total of seven questions are given in the Class 11 Maths chapter 1 exercise 1.5 that you can solve on your own. You can go through the class 11 maths chapter 1 exercise 1.5 solutions if you are facing problems while solving them.

**Also Read| **Sets Class 11th Notes

**Benefits of NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5:-**

- Venn's diagram is the foundation of all the properties of the complements of sets that you have come across class 11 maths chapter 1 exercise 1.5 chapter.
- Exercise 1.5 Class 11 Maths problems could be solved using Venn's diagram without using any property of complements of sets.
- You can try to prove all the laws of complements of sets using Venn's diagram. It will build your fundamentals of the set and its properties.

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Download EBook- Ex 1.5 class 11 developed by subject experts with a deep understanding of mathematical concepts and the CBSE curriculum.
- Class 11 maths ex 1.5 Solutions adhere to the latest CBSE syllabus, ensuring comprehensive coverage of relevant topics and concepts.
- Exercise 1.5 class 11 maths encompasses a wide variety of problems and questions, providing students with practice in different aspects of sets, thereby strengthening their foundation in advanced mathematics.
- Each class 11 ex 1.5 solution offers clear and concise explanations, aiding students in grasping the underlying concepts and honing their problem-solving skills.
- 11th class maths exercise 1.5 answers are frequently available online in various formats, including PDFs, providing convenient access for a broad spectrum of students.

Happy learning!!!

1. Let U = {1, 2, 3, 4, 5} and A = {1, 3} than Find A′ ?

A′ = U -A = { 2, 4 }

2. Let U = {1, 2, 3, 4, 5} and A = {1, 3} than Find (A′)' ?

A′ = (U -A)' = U - (U - A) = A = { 1, 3 }

3. Let U = { 1,2,3,4,5,7} and A = { 2,4,7} than find union of A and A' ?

A U A' = U = {1, 2,3,4,5,7}

4. What is the complement of universal set ?

The complement of the universal set is an empty set.

5. What is the complement of empty set ?

The complement of the empty set is a universal set.

6. Let U = { a, b, c, d, e, f, g, h}, and A = { b,c,d} than find A' ?

A' = U - A = { a, e, f, g, h}

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