ML AGGARWAL CLASS 12 Chapter 1 Relations and Functions Exercise 1.1

 Exercise 1.1

Question 1

(i) A={12,3,4,...10}
R={(x,y) : 2x-y=0}

Sol :

2x-y=0

2x=y

So, when x=1 , y=2 and so on (given in table below)

x	1	2	3	4	5
y	2	4	6	8	10

R={(1,2),(2,4),(3,6),(4,8),(5,10)}

Reflexivity 

Here, 

(1,1),(2,2),(3,3)....(10,10)∉R

So, R is not reflexive 

Symmetry

Here,

(1,2)∈R but (2,1)∉R

So, R is not symmetric

Transitivity

Here,

(1,2)∈R and (2,4)∈R but (1,4)∉R

So, R is not transitive

(ii) A=Z

R={(x,y) : x-y is an integer}

Sol :

Reflexivity :

∀ x∈Z

∵x-x=0 ∈Z

⇒(x,x)∈R

∴R is reflexive

Symmetric :

∀ x,y∈Z , (x,y)∈R

⇒x-y∈Z

⇒-(y-x)∈Z

⇒y-x ∈Z

⇒yRx

∵xRy⇒yRx

∴R is symmetric

Transitivity :

∀ x,y,t ∈Z , (x,y)∈R and (y,t)∈R

⇒x-y∈Z , y-z∈Z

⇒x-y=k ,k∈Z ...(i)

⇒y-z=k' k'∈Z...(ii)

Adding (i) and (ii)

⇒x-y+y-z=k+k'

⇒x-y=k+k'

⇒x-y=k'' , k''∈Z

⇒xRt

∵xRy , yRt ⇒ xRt

∴R is transitive

Question 2

If the relation R in the set A , where A={1,2,3,4,5,6} is defined by R={(x,y):y is divisible by x}, then express R in the roaster form. Also, determine whether the relation R is (i) Reflexive (ii) Symmetric (iii) Transitive 

Sol :

A={1,2,3,4,5,6}

R={(x,y) : y is divisible by x}

R={(1,1),(1,2),(2,2),(1,3),(3,3),(1,4),(2,4),(4,4),(1,5),(5,5),(1,6),(2,6),(3,6),(6,6)}

Reflexivity :

∵xRx ∀ x∈A

∴R is reflexive

Symmetric :

∵(1,2)∈R but (2,1)∉R

∴R is not symmetric

Transitivity :

∵xRy and yRz

⇒xRz

∴R is transitive

Question 3

If R is the relation defined on the set of natural numbers N as follows:

R={(x,y) : x,∈y N , 2x+y=41}

Find the domain and the range of the relation R

Determine whether the relation is reflexive , symmetric and transitive

Sol :

2x+y=41

y=41-2x

R={(1,39),(2,37),(3,35),(4,33),(5,31),(6,24),(7,27),(8,25),(9,23),(10,21),(11,19),(12,17),(13,15),(14,13),(15,11),(16,9),(17,7),(18,5),(19,3),(20,1)}

Domain of R={1,2,3,4,5......}

Range of R={39,38,37...1}

Reflexivity :

∵(1,1)∉R

∴R is not reflexive

Symmetric :

∵(1,39)∈R but (39,1)∉R

∴R is not symmetric

Transitivity :

∵(11,19)∈R and (19,3)∈R but (11,3)∉R

∴R is not transitive 

Question 4

Determine whether each of the following relations in the set A of human beings in a city at a particular time are reflexive , symmetric and transitive

(i) R={(x,y) : x and y work at the same place} [NCERT]

(ii) R={(x,y) : x and y live in the same locality} [NCERT]

(iii) R={(x,y) : x is exactly 7 cm taller than y} [NCERT]

(iv) R={(x,y) : x and y live within 2 kilometres} [NCERT]

Sol :

working..