Question

9. Show that the relation R - {(x, y): Ix - yl is divisible by 5) is an equivalence relation
in the set A - x x EZ and 0 sxs 12). Find the equivalence 171
(3)

Answer 1

Answer:

answer

Step-by-step explanation:

nswer

We know that a relation R is an equivalence relation, if it is reflexive, symmetric and transitive.

For reflexive:

∣a−a∣=0⇔(a,a)∈R, ∀a∈A

For symmetric:

(a,b)∈R⇔∣a−b∣=4k=∣b−a∣, k∈Z⇔(b,a)∈R, ∀(a,b)∈R

For transitive:

Let (a,b)∈R and (b,c)∈R

Thus, we have

(a,b)∈R⇔∣a−b∣=4k

1

, k

1

∈Z and

(b,c)∈R⇔∣b−c∣=4k

2

, k

2

∈Z

Since a, b and c are integers, we have

∣a−c∣=∣a−b+b−c∣=∣a−b∣±∣b−c∣=4(k

1

±k

2

)=4m, m∈Z⇔(a,c)∈R

So, we have shown that the relation R is reflexive, symmetric and transitive. Therefore, the relation is an equivalence relation.

Let x be the element of A such that (x,1)∈R

∣x−1∣ is a multiple of 4

⟹∣x−1∣=0,4,8,12

x−1=0,4,8,12

x=1,5,9 ( As 13∈

/

set of A)

Set of elements related to 1=1,5,9

Now, let us find the elements in the equivalence class [2].

Let y be the element of A such that (y,2)∈R

∣y−2∣ is a multiple of 4

⟹∣y−2∣=0,4,8,12

y−2=0,4,8,12

y=2,6,10 ( As 14∈

/

set of A)

Thus, the elements in the equivalence class [2]=2,6,10.