ML AGGARWAL CLASS 8 CHAPTER 6 Operation on Sets Venn Diagrams Excercise 6.1
Exercise 6.1
Question 1
Sol :
A={0,1,2,3,4,5,6,7,8} B={3,5,7,9,11}, C={0,5,10,20}
(i) A∪B={0,1,2,3,4,5,6,7,8,9,11}
(ii) A∪C={0,1,2,3,4,5,6,7,8,10,20}
(iii) B∪C={0,3,5,7,9,10,11,20}
(iv) A∩B={3,5,7}
(v) A∩C={0,5}
(vi) B∩C={5}
As B⋃C has 8 elements, n(B∪C)=8
As A∩B has 3 elements, n(A∩B)=3
As A∩C has 2 elements, n(A∩C)=2
As B∩C has 1 element, η(B∩C)=1
Question 2
Sol :
(i)
A={0,1,4,7} and $\xi=\{x \mid x \in w, x \leq 10\}$
Given ξ={0,1,2,3,4,5,6,7,8,9,10}
Complement of A=A′={2,3,5,6,8,9,10}
(ii)
A={ Consonants } and ξ={ alphabets of English }
Complement of A=A′, {vowels}
={a, e, i, o,u}
(iii) A = {boys in class viii of all schools in bengaluru} and
ξ = {student in class viii of all school in bengaluru}
Complement of A=A′, {girls in class viii of all school in bengaluru}
(iv) A={ letter of KAlKA} and ξ={ letters of KOlKATA}
Complement of A=A′={0,T}
(v) A={ odd natural numbers\} and ξ={ Whole numbers\}
Complement of A=A′={0,2,4,6,8,10,12…⋯−….}
Question 3
Sol :
A={x:x∈N and 3<x<7} and B={x:x∈ω and x≤4}
A={4,5,6} and B={0,1,2,3,4}
(i) A∪B={0,1,2,3,4,5,6}
(ii) A∩B={4}
(iii) A-B={5,6}$
A={x:x∈N and 3<x<7} and B={x:x∈ω and x≤4}
A={4,5,6} and B={0,1,2,3,4}
(i) A∪B={0,1,2,3,4,5,6}
(ii) A∩B={4}
(iii) A-B={5,6}$
in B-A=\{0,1,2,3}
Question 4
Sol :
p={0,1,2,3,4,5}Q={4,5,6,7,8}
(i) p∪Q={0,1,2,3,4,5,6,7,1}
(ii) P∩Q={4,5}
(iii) P-Q={0,1,2,3}
(iv) Q-P={6,7,8}
yes p∪Q is a proper superset of p∪Q but vice versa is not possible
since A contains elements not in B.
Question 5
Sol :
A={ letters of ward INTEGRITY\} B={ letters of word ReckonING}
(i) A∪B={I,N,T,E,G,R,Y,C,0}
(ii) A∩B={I,N,E,G,R}
(iii) A-B={T, Y}
(iv) B-A={c, k, 0}
(a) n(A)=7η(B)=8n(A∩B)=5n(A∪B)=10
n(A−B)=2n(B−A)=3
n(A)+n(B)−n(A∩B):7+8−5=10=η(A∪B)
(b) η(A∪B)−n(B)=10−8=2=n(A−B)
η(A)−n(A∩B)=7−5=2=n(A−B)
(c) η(A∪B)−n(A)=10−7=3=η(B−A)
η(B)−n(A∩B)=8−5=3=η(B−A)
d) η(A−B)+n(B−A)+n(A∩B)=2+3+5=10=n(A∪B)
Question 6
Sol :
ξ={10,11,12,13,14 .. 40}
A={5,10,15,20,2530,35,40}
B={6,12,18,24,30,36}....(3)
(i) A∪B={5,6,10,12,15,18,20,24,25,30,35,40}
A∩B={30}
(i) η(A)=8,n(B)=6,n(A∩B)=1η(A∪B)=13
η(A)+η(B)−n(A∩B)=8+6−1:13=n(A∪B)
Question 7
Sol :
(i) A′={5,9}
(ii) B′={1,2,3,5,7,9}
(iii) A∪B={1,2,3,4,6,7,8}
(iv) A∩B={4,6,8}
(v) A−B=A∩B′={1,2,3,7,}
(vi) B−A=B∩A′={3
(ii) (A∩B)′={1,2,3,5,7,9}
(viii) A′∪B′={1,2,3,5,7,9}
(a) (A∩B)′=A′∪B′={1,2,3,5,7,9} verified
(b) n(A)=7η(A′)=2η(ξ)=9
n(A)+n\left(A^{\prime}\right)=7+2=9=n(\xi) \text { verified }
(c)n(A∩B)+n((A∩B)′)
n(A∩B)=3;n((A∩B)′)=6
6+3=9=η(ξ). verified
(d) n(A−B)=4η(B−A)=0n(A∩B)=3
η(A−B)+η(B−A)+n(A∩B)=4+0+3=7=n(A∪B)
Question 8
Sol :
ξ1={x:x∈w,x≤10},A={x:x≥5}B={x:3≤x<8}
ξ={0,1,2,3,4,5,6,7,8,9,10}A={5,6,7,8,9,10}
B={3,4,5,6,7}
(i)
A∪B={3,4,5,6,7,8,9,10}A′={0,1,2,3,4}(A∪B)′={0,1,2}B′:{0,1,2,8;9,10}A′∩B′={0,1,2}
∴ (A∪B)′=A′∩B′={0,1,2}
(ii)
A∩B={5,6,7}−1(A∩B)′={0,1,2,3,4,8,9,10}
A′∪B′={0,1,2,3,4,8,9,10}
∴ (A∩B)′=A′UB′
(iii)
A−B={8,9,10}A∩B′={8,9,10}∴A−B=A∩B′
(iv)
B−A={3,4}B∩A′={3,4}∴B−A=B∩A′
Question 9
Sol :
n(A)=20,n(B)=16,n(A∪B)=30n(A∩B)=?
we know n(A∪B)=n(A)+n(B)−n(A∩B)
30=20+16−n(A∩B)
n(A∩B)=36−30=6n(A∩B)=C
n(A∩B)=c
Question 10
Sol :
n(5)=20:n(A′)=7n(A)=?
We know n(A)+n(A′)=n(ξ1)
n(A)=20-7=13
n(A)=13
n(A)=13
Question 11
Sol :
n(ξ1)=40n(A)=20n(B′)=16n(A∪B)=32
n(B)+n(B′)=n(ξ)⇒n(B)=40−16=24.n(B)=24
n(A∪B)=D(A)+η(B)−D(A∩B)
32=20+24−n(A∩B)
n(A∩B)=44−32=12
n(B)=24;n(A∩B)=12
Question 12
Sol :
n(ξ1)=32,n(A)=20,n(B)=16,n((A∪B)′)=4
(i) n(A∪B)=η(ξ)−n(A∪B)′)=32−4=28
n(A∪B)=28
(ii) n(A∩B)=n(A)+n(B)−n(A∪B)
20+28-28
36-28=8
n(A∩B)=8
(iii) n(A−B)=n(A)−n(A∩B)=20−8=12n(A−B)=12
Question 13
Sol :
η(ξ)=40:n(A′)=15,n(B)=12n((A∩B)′)=32
(i) n(A)=n(ξ)−n(A)=40−15=25
(ii) n(B′)=n(Σ1)−n(B)=40−12=28
(iii) n(A∩B)=n(ξ)−n((A∩B)′)=40−12=8
(iv) n(A∪B)=n(A)+n(B)−n(A∩B)=25+12−8=29
(v) n(A−B)=n(A)−n(A∩B)=25−8=17
(vi) η(B−A)=n(B)−n(A∩B)=12−8=4
Question 14
Sol :
η(A−B)=12,n(B−A)=16,n(A∩B)=5
(i) n(A) ii n(B) iii n(A∪B)
(i)n(A−B)=n(A)−n(A∩B)n(A)=12+5=17
(ii) n(B−A)=n(B)−n(A∩B)η(B)=16+5=21
(iii) n(A∪B)=n(A)+h(B)−n(A∩B)=17+21−5=38−5=33n(A∪B)=33
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