# Exercise 2.1

Question 1

(i) If find the values of and

Sol :

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

and

and

and b = 1

(ii) If find the values of x and y

Sol :

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

and

x = 2 and y=3

x = 2 and y=3

Question 2

If the ordered pairs and belong to the set find the values of x and y .

Sol :

The ordered pairs and belong to the set

Thus , we have :

x = a and -1 = b such that

x = 1

Also  ,

5 = a and y = b such that

y = 7

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

Question 3

If and write the set of all ordered pairs (a, b) such that a+b=5

Sol :

Given:

and

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

Thus, possible ordered pairs (a, b) are such that a+b=5

Question 4

If and then form the set of all ordered pairs (a, b) such that a divides b and a<b .

Sol :

Given:
and
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

Question 5

If and find and

Sol :

Given :

and

Now,

Question 6

Let and . Find and show it graphically.

Sol :

Given:

and

Now,

={(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}

To represent 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.



graphically .

{graph image}

Question 7

If and what are and

Sol :

Given :

Now ,

= {(1,2),(1,4),(2,2),(2,4),(3,2),(3,4)}

= {(2,1),(2,2),(2,3),(4,1),(4,2),(4,3)}

= {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

= {(2,2),(2,4),(4,2),(4,4)}

We observe:

Question 8

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

Sol :

Given :

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

Thus , we have:

= 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, 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)}

= {(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, a),(a, a),(a, a),(a, a),(a, a),(a, a),(a, a),(a, a)}

Question 9

Let A and B be two sets. Show that the sets and 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, e),(a, f),(b, e),(b, f),(c, e),(c, f)}

= {(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, a),(a, f),(b, a),(b, f),(c, a),(c, f)}

= {(a, a),(a, b),(a, c),(f, a),(f, b),(f, c)}

Here, and have two elements in common.

Thus, and 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 and
If (x, 1) , (y, 2) , (z, 1) are in find A and B, where x, y, z are distinct elements.

Sol :

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

Also ,

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

Question 11

Let A = {1,2,3,4} and Write R explicitly.

Sol :

Given :

A = {1,2,3,4}

We know:

1 divides 1 , 2 , 3 and 4

2 divides 2 and 4

3 divides 3

4 divides 4

= {(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)}

Question 12

If A = {-1,1} , find

Sol :

Given:

A = {-1,1}

Thus, we have:

= {(-1,-1),(-1,1),(1,-1),(1,1)}

And ,

= {(-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 = {(m, n),(n, m)}

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

(iii) If A = (1,2) , B = {3,4} , then

Sol :

Question 14

Sol :

Question 15

Sol :