RD sharma class solution 11 chapter 1 Set

Exercise 2.1

Question 1

(i) If \left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right), find the values of a and b

Sol :

By the definition of equality of ordered pairs, we have:

\Rightarrow\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)

\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3} and \left(b-\frac{2}{3}\right)=\frac{1}{3}

\Rightarrow \frac{a}{3}=\frac{5}{3}-1 and b=\frac{1}{3}+\frac{2}{3}

\Rightarrow \frac{\alpha}{3}=\frac{2}{3} and b = 1

 

(ii) If (x+1,1)=(3, y-2), find the values of x and y

Sol :

By the definition of equality of ordered pairs, we have:

\Rightarrow(x+1)=3 and 1=(y-2)

 x = 2 and y=3

 x = 2 and y=3

 

Question 2

If the ordered pairs (x,-1) and (5, y) belong to the set\{(a, b) : b=2 a-3\}, find the values of x and y .

Sol :

The ordered pairs (x,-1) and (5, y) belong to the set\{(a, b) : b=2 a-3\},

Thus , we have :

x = a and -1 = b such that b=2 a-3

\therefore-1=2 x-3

2 x=3-1

x = 1

 

Also  ,

5 = a and y = b such that b=2a-3

\therefore y=2(5)-3

y=10-3

y = 7

Thus , we get : x = 1 and y = 7

 

Question 3

If a \in[-1,2,3,4,5] and b \in[0,3,6], write the set of all ordered pairs (a, b) such that a+b=5

Sol :

Given:

a \in[-1,2,3,4,5] and b \in[0,3,6]

-1+6=5 , 2+3=5 and 5+0=5

Thus, possible ordered pairs (a, b) are \{(-1,6),(2,3),(5,0)\} such that a+b=5

 

Question 4

If a \in[2,4,6,9] and b \in[4,6,18,27], then form the set of all ordered pairs (a, b) such that a divides b and a<b .

Sol :

Given:
a \in[2,4,6,9] and b \in[4,6,18,27]
Here,

2 divides 4 , 6 and 18 and 2 is less than all of them .

6 divides 6 and 18 and 6 is less than 18 . 

9 divides 18 and 27 and 9 is less than 18 and 27 .

Now , Set of all ordered pairs (a,b) such that a divides and 

a<b=\{(2,4),(2,6),(2,18),(6,18),(9,18),(9,27)\}

 

Question 5

If A=\{1,2\} and B=(1,3), find A \times B and B \times A

Sol :

Given :

A=\{1,2\} and B=(1,3)

Now,

A \times B=((1,1),(1,3),(2,1),(2,3)\}

B \times A=\{(1,1),(1,2),(3,1),(3,2)\}

 

Question 6

Let A=(1,2,3) and B=\{3,4\} . Find A \times B and show it graphically.

Sol :

Given:

A=(1,2,3) and B=(3,4)

Now,

A \times B={(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}

To represent A \times B graphically, follow the given steps:

(a) Draw two mutually perpendicular lines- one horizontal and one vertical.

(b) On the horizontal line, represent the elements of set A ; and on the vertical line, represent the elements of set B.

(c) Draw vertical dotted lines through points representing elements of set A on the
horizontal line and horizontal lines through points representing elements of set B on the
vertical line.

The points of intersection of these lines will represent

*** QuickLaTeX cannot compile formula:
A\timesB

*** Error message:
Undefined control sequence \timesB.
leading text: $A\timesB

graphically .

{graph image}

 

Question 7

If A=\{1,2,3\} and B=\{2,4\}, what are A \times B, B \times A, A \times A, B \times B and (A \times B) \cap(B \times A) ?

Sol :

Given :

Now ,

A \times B = {(1,2),(1,4),(2,2),(2,4),(3,2),(3,4)}

B \times A = {(2,1),(2,2),(2,3),(4,1),(4,2),(4,3)}

A \times A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

B \times B = {(2,2),(2,4),(4,2),(4,4)}

We observe:

(A \times B) \cap(B \times A)=((2,2)\}

 

Question 8

If A and B are two set having 3 elements in common. If n(A)=5 , n(B)=4 , find n(A \times B)
and n[(A \times B) \cap(B \times A)]

Sol :

Given :

n(A)=5 and n(B)=4

Thus , we have:

n(A\times B)=(5\times 4) = 20

A and B are two sets having 3 elements in common.

Now,

Let:

A=(a, a, a, b, c) and B=(a, a, a, d)

Thus, we have:

(A \times B) = {(a, a),(a, a),(a, a),(a, d),(a, a),(a, a),(a, a),(a, d),(a, a),(a, a),(a, a),(a, d),(b, a),(b, a),(b, a),(b, d),(c, a),(c, a),(c, a),(c, d)}

(B \times A) = {(a, a),(a, a),(a, a),(a, b),(a, c),(a, a),(a, a),(a, b),(a, c),(a, a),(a, a),(a, b),(a, c),(d, a),(d, a),(d, a),(d, b),(d, c)}

[(A \times B) \cap(B \times A)] = {(a, a),(a, a),(a, a),(a, a),(a, a),(a, a),(a, a),(a, a)}

\therefore n[(A \times B) \cap(B \times A)]=9

 

Question 9

Let A and B be two sets. Show that the sets A \times B and B \times A have elements in common iff the sets A and B have an elements in common.

Sol :

Case (i): Let

A = (a , b , c)

B = (e , f)

Now , we have :

A \times B = {(a, e),(a, f),(b, e),(b, f),(c, e),(c, f)}

B \times A = {(e, a),(e, b),(e, c),(f, a),(f, b),(f, c)}

Thus, they have no elements in common

Case (ii): Let:

A = (a ,  b ,  c)

B = (a , f)

Thus ,we have :

A \times B={(a, a),(a, f),(b, a),(b, f),(c, a),(c, f)}

B \times A = {(a, a),(a, b),(a, c),(f, a),(f, b),(f, c)}

Here, A \times B and B \times A have two elements in common.

Thus, A \times B and B \times A will have elements in common iff sets A and B have elements in common.

 

Question 10

Let A and B be two sets such that n(A)=3 and n(B)=2
If (x, 1) , (y, 2) , (z, 1) are in A \times B, find A and B, where x, y, z are distinct elements.

Sol :

A is the set of all first entries in ordered pairs in A \times B and B is the set of all second entries in ordered pairs in A \times B .

Also ,

n(A)=3 and n(B)=2
\therefore A=\{x, y, z\} and B=(1,2)

 

Question 11

Let A = {1,2,3,4} and R=\{(a, b) : a \in A, b \in A, a \text { divides } b\} . Write R explicitly.

Sol :

Given :

A = {1,2,3,4}

R=\{(a, b) : a \in A, b \in A, a \text { divides } b\}

We know:

1 divides 1 , 2 , 3 and 4 

2 divides 2 and 4

3 divides 3

4 divides 4

\therefore R = {(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)}

 

Question 12

If A = {-1,1} , find A \times A \times A

Sol :

Given:

A = {-1,1}

Thus, we have:

A \times A = {(-1,-1),(-1,1),(1,-1),(1,1)}

And ,

A \times A \times A = {(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,-1),(1,1,-1),(1,1,1 )}

 

Question 13

State whether each of the following statements are true or false. If the statements is false, re-write the given statements correctly:

(i) If P = {m, n} and Q = {n, m} , then P\times Q = {(m, n),(n, m)}

(ii) If A and B are non-empty sets, then A \times B is a non-empty set of ordered pairs (x,y) such that x \in B and y \in A

(iii) If A = (1,2) , B = {3,4} , then \mathrm{A} \times(\mathrm{B} \cap \phi)=\phi

Sol :

 

 

 

Question 14

Sol :

Question 15

Sol :

 

 

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