VMC VIQ Scholarship Test
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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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Scholarship Test: Vidyamandir Intellect Quest (VIQ)
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
(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:-
Happy learning!!!
A′ = U -A = { 2, 4 }
A′ = (U -A)' = U - (U - A) = A = { 1, 3 }
A U A' = U = {1, 2,3,4,5,7}
The complement of the universal set is an empty set.
The complement of the empty set is a universal set.
A' = U - A = { a, e, f, g, h}
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