# 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 *b *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. leading text: $A\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 :