Question -
Answer -
Given that m is said to be related to n if m and n are integers and m тИТ 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 тИИ Z : m тИТ n is divisible by 13}
Let us check these properties on R.
Reflexivity:
Let m be an arbitrary element of Z.
Then, m тИИ R
тЗТ m тИТ m = 0 = 0 ├Ч 13
тЗТ m тИТ m is divisible by 13
тЗТ (m, m) is reflexive on Z.
Symmetry:
Let (m, n) тИИ R.
Then, m тИТ n is divisible by 13
тЗТ m тИТ n = 13p
Here, p тИИ Z
тЗТ n тАУ m = 13 (тИТp)
Here, тИТp тИИ Z
тЗТ n тИТ m is divisible by 13
тЗТ (n, m) тИИ R for all m, n тИИ Z
So, R is symmetric on Z.
Transitivity:
Let (m, n) and (n, o) тИИR
тЗТ m тИТ n and n тИТ o are divisible by 13
тЗТ m тАУ n = 13p and n тИТ o = 13q for some p, q тИИ Z
Adding the above two equations, we get
m тАУ n + n тИТ o = 13p + 13q
тЗТ mтИТo = 13 (p + q)
Here, p + q тИИ Z
тЗТ m тИТ o is divisible by 13
тЗТ (m, o) тИИ R for all m, o тИИ Z
So, R is transitive on Z.
Therefore R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on Z.