Study Materials: ncert solutions

Our ncert solutions for Solutions 1. Sets - Exercise 1.5 | Class 11 Mathematics - Toppers Study is the best material for English Medium students cbse board and other state boards students.

 

Solutions ⇒ Class 11th ⇒ Mathematics ⇒ 1. Sets

Solutions 1. Sets - Exercise 1.5 | Class 11 Mathematics - Toppers Study

Topper Study classes prepares ncert solutions on practical base problems and comes out with the best result that helps the students and teachers as well as tutors and so many ecademic coaching classes that they need in practical life. Our ncert solutions for Solutions 1. Sets - Exercise 1.5 | Class 11 Mathematics - Toppers Study is the best material for English Medium students cbse board and other state boards students.

1. Sets

| Exercise 1.5 |

Solutions 1. Sets - Exercise 1.5 | Class 11 Mathematics - Toppers Study


Exercise 1.5 


Q1. Let 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 }. Find

(i) A′

(ii) B′

(iii) (A ∪ C)′

(iv) (A ∪ B)′

(v) (A′)′

(vi) (B – C)′

Solution: Given that

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' = {5, 6, 7, 8, 9}

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

(iii) A ∪ C = {1, 2, 3, 4, 5, 6}

Therefore, (A ∪ C)′ = {7, 8, 9}

(iv) A ∪ B = {1, 2, 3, 4, 6, 8}

Therefore, (A ∪ B)′ = {5, 7, 9}

(v) A' = {5, 6, 7, 8, 9}

(A')' = A = {1, 2, 3, 4} 

(vi) B - C = {2, 8}

   (B - C)' = 1, 3, 4, 5, 6, 7, 9} 

Q2. 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}

Solution: Given that 

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

(i) A = {a, b, c} 

   A' = {d, e, f, g, h}

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

    B' = {a, b, c, h}

(iii) C = {a, c, e, g} 

    C' = {b, d, f, h} 

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

    D' = {b, c, d e}

Q3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(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}
(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 }

Solution: Given that U = { 1, 2, 3, 4, 5, 6, 7 ....}

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

Or A = {2, 4, 6, 8 .....} 

A' = { 1, 3, 5, 7 .....}

   = {x : x is an odd natural number}

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

Or     B = { 1, 3, 5, 7 .....} 

B' = {2, 4, 6, 8 .....} 

   = {x : x is an even natural number} 

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

Or     C = {3, 6, 9 ....} 

C' = {1, 2, 4, 5, 7, 8, 10 .....}

   = {x: x N and x is not a multiple of 3}

(iv) Let D = { x : x is a prime number }

Or     D = {2, 3, 5, 7, 11 ... }

D' = {1, 4, 6, 8, 9, 10 ...... } 

   = {x: x is a positive composite number and x = 1}

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

Or     E = {15, 30, 45 .....}

E' = {x: x is a natural number that is not divisible by 3 or 5}

(vi) Let F = { x : x is a perfect square } 

F' = {x: x N and x is not a perfect square}

(vii) Let G = {x: x is a perfect cube}

G' = {x: x N and x is not a perfect cube}

(viii) Let H = {x: x + 5 = 8}

H' = {x: x N and x ≠ 3}

(ix) Let I = {x: 2x + 5 = 9}

I' = {x: x N and x ≠ 2}

(x) Let J = {x: x ≥ 7}

J' = {x: x N and x < 7}

(xi) Let K = {x: x N and 2x + 1 > 10}

K = {x: x N and x ≤ 9/2}

Q4. 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′

Solution: 

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

(A ∪ B)′ = A′ ∩ B′

A ∪ B = {2, 3, 4, 5, 6, 7, 8} 

LHS = (A ∪ B)′ = {1, 9} ...(i)

RHS = A′ ∩ B′

= {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} 

= {1, 9} .... (ii) 

LHS = RHS 

Hence Verified.

Solution:

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

(A ∩ B)′ = A′ ∪ B′

A ∩ B = {2}

LHS = (A ∩ B)′ = {1, 3, 4, 5, 6, 7, 8, 9 }

RHS = A′ ∪ B′

{1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} 

= {1, 3, 4, 5, 6, 7, 8, 9 }

LHS = RHS 

Hence Verified 

Q5. Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′,

(ii) A′ ∩ B′,

(iii) (A ∩ B)′,

(iv) A′ ∪ B′

Solution:

(i) (A ∪ B)′

Venn diagram of (A ∪ B)′

(ii) A′ ∩ B′,

Venn diagram of A′ ∩ B′

Note: Venn diagram of A′ ∩ B′ will be same as (A ∪ B)′

Because (A ∪ B)′ = A′ ∩ B′

(iii) (A ∩ B)′

Venn diagram of (A ∩ B)′

(iv) A′ ∪ B′

Venn diagram of A′ ∪ B′

Q6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?

Solution: 

A = {the set of all triangles with at least one angle different from 60°}

A' = {the set of all equilateral triangles}

Q7. Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . .

(ii) φ′ ∩ A = . . .

(iii) A ∩ A′ = . . .

(iv) U′ ∩ A = . . .

Solution: 

(i) A ∪ A′ = U

(ii) φ′ = U 

Therefore φ′ ∩ A = U ∩ A = A 

so, φ′ ∩ A = A 

(iii) A ∩ A′ = φ

(iv) U′ ∩ A = φ

Select Your CBSE Classes

Important Study materials for classes 06, 07, 08,09,10, 11 and 12. Like CBSE Notes, Notes for Science, Notes for maths, Notes for Social Science, Notes for Accountancy, Notes for Economics, Notes for political Science, Noes for History, Notes For Bussiness Study, Physical Educations, Sample Papers, Test Papers, Mock Test Papers, Support Materials and Books.

 

 

Mathematics Class - 11th

NCERT Maths book for CBSE Students.

books

Study Materials List:

Solutions ⇒ Class 11th ⇒ Mathematics
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle Of Mathematical Induction
5. Complex Numbers and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability

Topper's Study

New Books