Question

a relation R on a collection of set of integers defined by R= {(x,y) : x-y is a multiple of 3} . show that R is an equivalence relation on Z​

Answer 1

Answer:

R={(x,y):x−yis an integer}

Now, for every x∈Z,(x,x)∈R as x−x=0 is an integer.

∴R is reflexive.

Now, for every x,y∈Z if (x,y)∈R, then x−y is an integer.

⇒−(x−y) is also an integer.

⇒(y−x) is an integer.

∴(y,x)∈R

⇒R is symmetric.

Now,

Let (x,y) and (y,z)∈R, where x,y,z∈Z.

⇒(x−y) and (y−z) are integers.

⇒x−z=(x−y)+(y−z) is an integer.

∴(x,z)∈R

∴R is transitive.

Hence, R is reflexive, symmetric, and transitive.