Given as m is said to be related to n if m and n are integers and m − n is divisible by 13
Let us check whether the given relation is equivalence or not.
To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.
Let R = {(m, n): m, n ∈ Z : m − n is divisible by 13}
We have to check these properties on R.
Reflexivity:
Let m be an arbitrary element of Z.
Then, m ∈ R
⇒ m − m = 0 = 0 × 13
⇒ m − m is divisible by 13
⇒ (m, m) is reflexive on Z.
Symmetry:
Let (m, n) ∈ R.
Then, m − n is divisible by 13
⇒ m − n = 13p
Here, p ∈ Z
⇒ n – m = 13(−p)
Here, −p ∈ Z
⇒ n − m is divisible by 13
⇒ (n, m) ∈ R for all m, n ∈ Z
Therefore, R is symmetric on Z.
Transitivity:
Let (m, n) and (n, o) ∈ R
⇒ m − n and n − o are divisible by 13
⇒ m – n = 13p and n − o = 13q for some p, q ∈ Z
Adding above two equations, we get
m – n + n − o = 13p + 13q
⇒ m − o = 13(p + q)
Here, p + q ∈ Z
⇒ m − o is divisible by 13
⇒ (m, o) ∈ R for all m, o ∈ Z
So, R is transitive on Z.
∴ R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on Z.
