Show that the relation S in the set given by S = {(a, b) : a, b ∈ Z, |a - b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.

To show an equivalence relation , We need to check for all Reflexivity, Symmetric and Transitivity

For Reflexivity:

Let a∈Z

Since, |a - a| = 0 is divisible by 4

Then (a, a)∈S

Therefore, S is reflexive

For Symmetric:

Let (a, b)∈S

Since, |a - b| is divisible by 4

And, |b - a| is also divisible by 4

So, (b, a)∈S

Hence, S is symmetric

For Transitivity:

Let (a, b)∈S and (b, c)∈S

Since, |a - b| and |b - c| are divisible by 4

And, |a - b| = 4p and |b - c| = 4q for some p, q∈Z

Then, |a - c| = |(a - b) + (b - c)| = 4(p + q)

|a - c| is divisible by 4

So, (a, c)∈S

As S is reflexive , symmetric and transitive its an equivalence relation.

Now,

Let (1, x) ∈ S , x∈A

x - 1 = 4p , for some p∈Z

z = 1 + 4p

if we put p = 1, 2 then

z = 5, 9

since, |1 - 0| is not divisible by 4, set of elements related to 1 is [5, 9]