Let `n` be a fixed positive integer. Define a relation `R` on `Z` as follows: `(a ,\ b) in RhArra-b` is divisible by `ndot` Show that `R` is an equivalence relation on `Zdot`

Answer

N/a

Explanation

Given that, ` AA a, b in Z,` aRb of and only if a - b is divisible by n.
Now,
I. Reflexive
aRa`implies`(a - a) is divisible by n, which is true for any integer a as 'O' is divisible by n.
Hence, R is reflexive.
II. Symmetric
aRb
`implies a - b` is divisible by n.
`implies -b+a` is divisible by n.
`implies -(b - a)` is divisible by n.
`implies (b-a)` is divisible by n.
`implies ` bRa
Hence, R is symmetric.
III. Transitive
Let aRb and bRc
`implies (a-b)` is divisible by n and `(b-c)` is divisible by n
` implies (a-b)+(b-c)` is divisibly by n
`implies (a-c)`is divisible by n
aRc
Hence, R is transitive.
So, R is an equivalence relation.