Question

R is the set of Real numbers and atb=ab.​

Answer 1

Answer:Solution: (1)

a ∈ R

a – a + √3 = √3 is an irrational number

(a, a) ∈ R

R is reflexive.

a, b ∈ R

a – b + √3 is an irrational number

b – a + √3 is an irrational number

(b, a) ∈ R

R is symmetric.

(a, b) and (b, c) ∈ R

a – b + √3 is an irrational number

b – c + √3 is an irrational number

On adding the above two expressions,

a – c + 2√3 is irrational.

(a, c) ∈ R

R is transitive.

So, R is an equivalence relation.

Step-by-step explanation:

Answer 2

Answer:

R={(a, b) : 3 divides a−b}

Case I : Reflexive

Since,  a−a=0

And                                              

3  divides  0,   ∴  20=0

⇒2 divides  a−a

∴(a, 0) ∈ R.

∴R is reflexive

Case II : - Symmetric

If 3 divides a - b,

then, 3 divides - (a - b) i.e. b - a

 Hence, If ( a , b) ∈R, Then ( b , a) ∈R

 ∴R is symmetric.

 Case III. Transitive.

 If 3 divides ( a - b) and 3 divides ( b - c)

 So, 2 divides ( a - b) + ( b - c)

 also,

 2 divides ( a - c)

 ∴If(a,b) ∈R and (b, c) ∈R

 then,

 (a, c) ∈R

 Thus, R is an equivalence relation in z.

Hence, this is the answer.