Rd sharma solution class 12 chapter Relations

Exercise 1.1

Question 1

Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:

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

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x \in R

\Rightarrow x and x work at the same place is true since they are same 

\Rightarrow (x,x)\in R

So, R is a reflexive relation

 

(Symmetry)

Let (x,y) \in R

\Rightarrow x and y work at the same place .

\Rightarrow y and x work at the same place .

\Rightarrow (y,x) \in R

So, R is symmetric relation

 

(Transitivity)

Let (x,y) \in R and (y,z) \in R . Then,

\Rightarrow x and y work at the same place .

\Rightarrow y and z work at the same place .

\Rightarrow x , y and z all work at the same place .

\Rightarrow (x,z) \in R 

So, R is a transitive relation.

 

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

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x \in R

\Rightarrow x and x live in the same locality is true since they are same.

\Rightarrow (x,x)\in R

So, R is a reflexive relation

 

(Symmetry)

Let (x,y) \in R

\Rightarrow x and y live in the same locality.

\Rightarrow y and x live in the same locality.

\Rightarrow (y,x) \in R

So, R is symmetric relation

 

(Transitivity)

Let (x,y) \in R and (y,z) \in R . Then,

\Rightarrow x and y live in the same locality.

\Rightarrow y and z live in the same locality.

\Rightarrow x , y and z live in the same locality.

\Rightarrow (x,z) \in R 

So, R is a transitive relation.

 

(iii) R = {(x , y) : x is wife of y }

Sol :

(Reflexivity)

Let x be an element of R . Then,

x is wife of x which cannot be true.

\Rightarrow (x,x) \in R 

So, R is not a reflexive relation

 

(Symmetry)

Let (x,y) \in R

\Rightarrow x is wife of y.

\Rightarrow x is female and y is male.

\Rightarrow y cannot be wife of x as y is husband of x.

\Rightarrow (y,x)\not\in R

So, R is not a symmetric relation.

 

(Transitivity)

Let (x,y) \in R and (y,z) \in R . Then,

\Rightarrow x is wife of y and y is husband of z 

which is a contradiction

\Rightarrow (x,z)\not\in R

So, R is not transitive 

 

(iv) R = {(x , y) : x is father of and y }

Sol :

(Reflexivity)

Let x be an arbitrary element of R . Then,

x is father of x which cannot be true since no one can be father of himself.

\Rightarrow (x,x) \not \in R 

So, R is not a reflexive relation

 

(Symmetry)

Let (x,y) \in R

\Rightarrow x is father of y.

\Rightarrow y is son or daughter of x .

\Rightarrow (y,x)\not\in R

So, R is not a symmetric relation.

 

(Transitivity)

Let (x,y) \in R and (y,z) \in R . Then,

\Rightarrow x is father of y and y is father of z 

\Rightarrow x is grandfather of z.

\Rightarrow (x,z)\not\in R

So, R is not transitive 

 

Question 2

Three relations R_1,R_2~and~R_3 are defined on a set A=\left\{a,b,c\right\} as follows:

R_1={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}

R_2=\left\{(a,a)\right\}

R_3=\left\{(b,c)\right\}

R_4={(a,b),(b,c),(c,a)}

Find whether or not each of the relations R_1,R_2,R_3,R_4 on A is (i) reflexive  (ii) symmetric (iii) transitive.

Sol :

(i) R_1={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}

Reflexive as (a,a)\in R_1(b,b)\in R_1 and (c,c)\in R_1

Not symmetric as (a,b)\in R_1 but (b,a)\not\in R_1

Not transitive as (b,c)\in R_1 and (c,a)\in R_1 but (b,a)\not\in R_1

 

(ii) R_2=\left\{(a,a)\right\}

Not Reflexive as (a,a)\in R_2 but (b,b)\not\in R_2 and (c,c)\not\in R_2

Symmetric as (a,a)\in R_2 and (a,a)\in R_2

Transitive as (a,a)\in R_2 and (a,a)\in R_2 also (a,a)\in R_2

 

 

(iii) R_3=\left\{(b,c)\right\}

Not Reflexive as (a,a)\not\in R_3 , (b,b)\not\in R_3 and (c,c)\not\in R_3

Not symmetric as (b,c)\in R_3 but (c,b)\not\in R_3

Transitive as R_3 has only two elements

 

 

(iv) R_4={(a,b),(b,c),(c,a)}

Not Reflexive as (a,a)\not\in R_4 , (b,b)\not\in R_4 and (c,c)\not\in R_4

Not symmetric as (a,b)\in R_4 but (b,a)\not\in R_4

Not transitive as (a,b)\in R_4 and (b,c)\in R_4 but (a,c)\not\in R_4

 

Question 3

Test whether the following relations R_1,R_2~and~R_3 are (i) reflexive  (ii) symmetric  and (iii) transitive

(i) R_1~on~Q_o defined by (a,b)\in R_1 \Leftrightarrow a=1/b

Sol :

(Reflexivity)

Let “a” be an arbitrary element of R_1.Then,

a\in R_1

\Rightarrow a\not=\dfrac{1}{a} for all a\in Q_o

So, R_1 is not reflexive.

 

(Symmetry)

Let (a,b) \in R_1 . Then,

\Rightarrow a=\dfrac{1}{b}

\Rightarrow b=\dfrac{1}{a}

\Rightarrow (b,a) \in R_1

So, R_1 is symmetric.

 

(Transitivity)

Let (a,b)\in R_1 and (b,c)\in R_1

\Rightarrow a=\dfrac{1}{b} and b=\dfrac{1}{c}

\Rightarrow a=\dfrac{1}{\bigg(\dfrac{1}{c}\bigg)}=c

\Rightarrow a\not=\dfrac{1}{c}

\Rightarrow (a,c)\not\in R_1

So, R_1 is not transitive

 

 

(ii) R_2~on~Z defined by (a,b)\in R_2 \Leftrightarrow |a-b|\leq5

Sol :

(Reflexivity)

Let “a” be an arbitrary element of R_2.Then,

a\in R_2

\Rightarrow |a-a|=0\leq5

So, R_2 is reflexive.

 

(Symmetry)

Let (a,b) \in R_2 . Then,

\Rightarrow |a-b|\leq5

\Rightarrow |b-a|\leq5 \because |a-b|=|b-a|

\Rightarrow (b,a) \in R_2

So, R_2 is symmetric.

 

(Transitivity)

Let (a,b)\in R_2 and (b,c)\in R_2

\Rightarrow |a-b|\leq5

\Rightarrow |b-c|\leq5

\Rightarrow but~|a-c|\geq5

\Rightarrow (a,c)\not\in R_2

So, R_2 is not transitive

For better illustration example is give

if~a=15,b=11,c=7

\Rightarrow |15-11|\leq5~and~|11-7|\leq5

but |15-7|\geq5

 

 

(iii) R_3~on~R defined by (a,b)\in R_3 \Leftrightarrow a^2-4ab+3b^2=0

Sol :

 

(Reflexivity)

Let “a” be an arbitrary element of R_3.Then,

a\in R_3

\Rightarrow a^2-4a\times a+3a^2=0

So, R_3 is reflexive.

 

(Symmetry)

Let (a,b) \in R_3 . Then,

\Rightarrow a^2-4ab+3b^2=0

\Rightarrow but~b^2-4ba+3a^2\not=0 for all (a,b)\in R

\Rightarrow (b,a)\not \in R_3

So, R_3 is not symmetric.

 

(Transitivity)

Let (a,b)\in R_3 and (b,c)\in R_3

(1,2)\in R_3 and (2,3)\in R_3

\Rightarrow 1-8+6=0 and 4-24+27=0

\Rightarrow but~(a,c)\not\in R_3

\Rightarrow but~1-12+9\not=0

So, R_3 is not transitive

 

Question 4

Let A={1,2,3} and let R_1={(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)} 

R_2={(2,2),(3,1),(1,3)}

R_3={(1,3),(33)}

Find whether or not each of the relations R_1,R_2,R_3 on A is (i) reflexive  (ii) symmetry  (iii) transitive

Sol :

(i) R_1={(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)}

Reflexive as (1,1)\in R_1(2,2)\in R_1 and (3,3)\in R_1

Symmetric as (2,1)\in R_1 but (1,2)\notin R_1

Not Transitive as (2,1)\in R_1 and (1,3)\in R_1 but (2,3)\notin R_1

 

(ii) R_2={(2,2),(3,1),(1,3)}

Not Reflexive as (1,1)\not\in R_2 and (3,3)\not\in R_2

Symmetric as (3,1)\in R_2 but (1,3)\notin R_2

Not Transitive as (3,1)\in R_2 and (1,3)\in R_2 but (3,3)\notin R_2

 

(iii) R_3={(1,3),(33)}

Not Reflexive as (1,1)\notin R_3 and (2,2)\notin R_3

Not Symmetric as (1,3)\in R_3 but (3,1)\notin R_3

Transitive as (1,3)\in R_3 and (3,3)\in R_3 also (1,3)\in R_3

 

 

Question 5

The following relations are defined on the set of real numbers .

(i) [latex]aRb \text{ if } a-b>0[latex]

(ii) [latex]aRb \text{ if } 1+ab>0[latex]

(iii) [latex]aRb \text{ if } |a|\leq b[latex]

Sol :

(i) [latex]aRb \text{ if } a-b>0[latex]

( Reflexivity )

 

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