Question

Determine whether the following relation is reflexive,summetric or transitive ::
Relation R in the set Z of all integers defined by R={(x,y):x-y is an integer}

Answer 1

Relation R is reflexive, symmetric as well as transitive.
Reason

Reflexive
Let x belongs to Z
x - x is an integer.
Therefore, (x,x) belongs to R
Hence R is reflexive.

Symmetric
Let x,y belongs to Z
Such that (x,y) belongs to R
x-y is an integer
Therefore, y - x is also an integer
Hence, (y,x) belongs to R.
Hence, R is symmetric

Transitive
Let x,y,z belongs to Z
Such that (x,y) and (y,z) belongs to R
It implies, x-y is an integer and y-z is an integer.

Therefore, x-y+y-z is an integer
It means x-z is an integer.
Hence, (x,z) belongs to R.
Hence, R is transitive