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) $A \cup B=\{0,1,2,3,4,5,6,7,8,9,11\}$

(ii) $A \cup C=\{0,1,2,3,4,5,6,7,8,10,20\}$

(iii) $B \cup C=\{0,3,5,7,9,10,11,20\}$

(iv) $A \cap B=\{3,5,7\}$

(v) $A \cap C=\{0,5\}$

(vi) $B \cap C=\{5\}$

As B⋃C has 8 elements, $n(B \cup C)=8$

As $A \cap B$ has 3 elements, $n(A \cap B)=3$

As $A \cap C$ has 2 elements, $n(A \cap C)=2$

As $B \cap C$ has 1 element, $\eta(B \cap C)=1$


Question 2

Sol :

(i)

$A=\{0,1,4,7\}$ and $\xi=\{x \mid x \in w, x \leq 10\}$

Given $\xi_=\{0,1,2,3,4,5,6,7,8,9,10\}$

Complement of $A=A^{\prime}=\{2,3,5,6,8,9,10\}$


(ii)

$A=\{$ Consonants $\}$ and $\xi=\{$ alphabets of English $\}$

Complement of $A=A^{\prime}$, {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 $A=A^{\prime}$, {girls in class viii of all school in bengaluru}


(iv) $A=\{$ letter of $\mathrm{KAlKA}\}$ and $\xi=\{$ letters of $\mathrm{KOlKATA}\}$

Complement of $A=A^{\prime}=\{0, T\}$


(v) $A=\{$ odd natural numbers\} and $\xi=\{$ Whole numbers\}

Complement of $A=A^{\prime}=\{0,2,4,6,8,10,12 \ldots \cdots-\ldots .\}$

 

Question 3

Sol :

$A=\{x: x \in N$ and $3<x<7\}$ and $B=\{x: x \in \omega$ and $x \leq 4\}$

$A=\{4,5,6\} \quad$ and $B=\{0,1,2,3,4\}$

(i) $A \cup B=\{0,1,2,3,4,5,6\}$

(ii) $A \cap B=\{4\}$

(iii) A-B={5,6}$

in B-A=\{0,1,2,3}


Question 4

Sol :

$p=\{0,1,2,3,4,5\} \quad Q=\{4,5,6,7,8\}$

(i) $p \cup Q=\{0,1,2,3,4,5,6,7,1\}$

(ii) $P \cap Q=\{4,5\}$

(iii) P-Q={0,1,2,3}

(iv) Q-P={6,7,8}

yes  $p \cup Q$ is a proper superset of  $p \cup Q$ but vice versa is not possible 
since A contains elements not in B.



Question 5

Sol :

$A=\left\{\right.$ letters of ward INTEGRITY\} $B=\left\{\begin{array}{l}\text { letters of word } \\ \text { ReckonING\} }\end{array}\right.$

(i) $A \cup B=\{I, N, T, E, G, R,  Y, C, 0\}$

(ii) $A \cap B=\{I, N, E, G, R\}$

(iii) A-B={T, Y}

(iv) B-A={c, k, 0}


(a) $n(A)=7 \quad \eta(B)=8 \quad n(A \cap B)=5 \quad n(A \cup B)=10$

$n(A-B)=2 \quad n(B-A)=3$

$n(A)+n(B)-n(A \cap B): 7+8-5=10=\eta(A \cup B)$


(b) $\eta(A \cup B)-n(B)=10-8=2=n(A-B)$

$\quad \eta(A)-n(A \cap B)=7-5=2=n(A-B)$


(c) $\eta(A \cup B)-n(A)=10-7=3=\eta(B-A)$

$\eta(B)-n(A \cap B)=8-5=3=\eta(B-A)$


d) $\eta(A-B)+n(B-A)+n(A \cap B)=2+3+5=10=n(A \cup B)$



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) $A \cup B=\{5,6,10,12,15,18,20,24,25,30,35,40\}$

$A \cap B=\{30\}$

(i) $\eta(A)=8, n(B)=6, n(A \cap B)=1 \quad \eta(A \cup B)=13$

$\eta(A)+\eta(B)-n(A \cap B)=8+6-1: 13=n(A \cup B)$


Question 7

Sol :

(i) $A^{\prime}=\{5,9\}$

(ii) $B^{\prime}=\{1,2,3,5,7,9\}$

(iii) $A \cup B=\{1,2,3,4,6,7,8\}$

(iv) $A \cap B=\{4,6,8\}$

(v) $A-B=A \cap B^{\prime}=\{1,2,3,7,\}$

(vi) $B-A=B \cap A^{\prime}=\{3$

(ii) $(A \cap B)^{\prime}=\{1,2,3,5,7,9\}$

(viii) $A^{\prime} \cup B^{\prime}=\{1,2,3,5,7,9\}$


(a) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}=\{1,2,3,5,7,9\}$ verified

(b) $n(A)=7 \quad \eta\left(A^{\prime}\right)=2 \quad \eta(\xi)=9$

 n(A)+n\left(A^{\prime}\right)=7+2=9=n(\xi) \text { verified }


(c)$n(A \cap B)+n\left((A \cap B)^{\prime}\right)$

$n(A \cap B)=3 ; n\left((A \cap B)^{\prime}\right)=6$

$6+3=9=\eta(\xi) .$ verified


(d) $n(A-B)=4 \quad \eta(B-A)=0 \quad n(A \cap B)=3$

$\quad \eta(A-B)+\eta(B-A)+n(A \cap B)=4+0+3=7=n(A \cup B)$



Question 8

Sol :

$\xi_{1}=\{x: x \in w, x \leq 10\}, A=\{x: x \geq 5\} \quad B=\{x: 3 \leq x<8\}$

$\xi=\{0,1,2,3,4,5,6,7,8,9,10\} \quad A=\{5,6,7,8,9,10\}$

$B=\{3,4,5,6,7\}$

(i)
 $\begin{aligned} A \cup B=\{3,4,5,6,7,8,9,10\} & A^{\prime}=\{0,1,2,3,4\} \\(A \cup B)^{\prime}=\{0,1,2\} & B^{\prime}:\{0,1,2,8 ; 9,10\} \\ A^{\prime} \cap B^{\prime}=\{0,1,2\} \end{aligned}$

 ∴ $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}=\{0,1,2\}$

(ii)
 $A \cap B=\{5,6,7\}-1(A \cap B)^{\prime}=\{0,1,2,3,4,8,9,10\}$
$A^{\prime} \cup B^{\prime}=\{0,1,2,3,4,8,9,10\}$

 ∴ $(A \cap B)^{\prime}=A^{\prime} U B^{\prime}$

(iii) 
$\begin{aligned} A-B &=\{8,9,10\} \\ A \cap B^{\prime} &=\{8,9,10\} \end{aligned} \quad \therefore A-B=A \cap B^{\prime}$

(iv) 
$\begin{aligned} B-A &=\{3,4\} \\ B \cap A^{\prime} &=\{3,4\} \end{aligned} \quad \therefore \quad B-A=B \cap A^{\prime}$




Question 9

Sol :

$n (A)=20, \quad n (B)=16, n(A \cup B)=30 \quad n (A \cap B)=?$

we know $n (A \cup B)= n (A)+n(B)-n (A \cap B)$

$30=20+16-n (A \cap B)$

$\begin{aligned} n (A \cap B)=36-30=6 \\ n (A \cap B)=C \end{aligned}$

$n (A \cap B)=c$


Question 10

Sol :

$n(5)=20: n\left(A^{\prime}\right)=7 \quad n(A)=?$

We know $\quad n(A)+n\left(A^{\prime}\right)=n\left(\xi_{1}\right)$

n(A)=20-7=13

n(A)=13


Question 11

Sol :

$n\left(\xi_{1}\right)=40 \quad n(A)=20 \quad n\left(B^{\prime}\right)=16 \quad n(A \cup B)=32$

$\begin{aligned} n(B)+n\left(B^{\prime}\right)= n (\xi) \Rightarrow & n(B)=40-16=24 . \\ & n(B)=24 \end{aligned}$

$n (A \cup B)=D(A)+\eta(B)-D(A \cap B)$
$32=20+24-n(A \cap B)$
$n(A \cap B)=44-32=12$
$n(B)=24 ; n(A \cap B)=12$


Question 12

Sol :

$n\left(\xi_{1}\right)=32, \quad n(A)=20, n(B)=16, n\left((A \cup B)^{\prime}\right)=4$

(i) $\begin{aligned} n(A \cup B) &\left.=\eta(\xi)-n(A \cup B)^{\prime}\right) \\ &=32-4=28 \end{aligned}$

$n(A \cup B)=28$


(ii) $n(A \cap B)=n(A)+n(B)-n(A \cup B)$

20+28-28

36-28=8

$n(A \cap B)=8$


(iii) $\begin{aligned} n(A-B) &=n(A)-n(A \cap B) \\ &=20-8=12 \\ & n(A-B)=12 \end{aligned}$




Question 13

Sol :

$\eta(\xi)=40 \quad: n\left(A^{\prime}\right)=15, n(B)=12 \quad n\left((A \cap B)^{\prime}\right)=32$

(i) $n(A)=n(\xi)-n(A)=40-15=25$


(ii) $n\left(B^{\prime}\right)=n\left(\Sigma_{1}\right)-n(B)=40-12=28$


(iii) $n (A \cap B)=n(\xi)-n\left((A \cap B)^{\prime}\right)=40-12=8$


(iv) $n(A \cup B)=n(A)+n(B)-n(A \cap B)=25+12-8=29$

(v) $n(A-B)=n(A)-n(A \cap B)=25-8=17$

(vi) $\eta(B-A)=n(B)-n(A \cap B)=12-8=4$



Question 14

Sol :

$\eta(A-B)=12, n(B-A)=16, \quad n(A \cap B)=5$

(i) $n(A) \quad$ ii $\quad n(B)$ iii $n(A \cup B)$


(i)$\begin{aligned} n(A-B) &=n(A)-n(A \cap B) \\ n(A) &=12+5=17 \end{aligned}$


(ii) $\begin{aligned} n(B-A) &=n(B)-n(A \cap B) \\ \eta(B) &=16+5=21 \end{aligned}$


(iii) $\begin{aligned} n(A \cup B) &=n(A)+h(B)-n(A \cap B) \\ &=17+21-5=38-5=33 \\ & n(A \cup B)=33 \end{aligned}$

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