Question -
Answer -
Given that m is said to be related to n if m and n are integers and m − n is divisible by 13
Now we have to check whether the given relation is equivalence or not.
To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.
Let R = {(m, n): m, n ∈ Z : m − n is divisible by 13}
Let us 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
So, 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 the 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.
Therefore R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on Z.