The relation R on Z is defined by R Colon Equals a bEpsilon

The relation R on Z is defined by R Colon Equals {(a, b)Epsilon Z x Z: a^2 Strictly Equivalent To b^2 (mod 3)}. Prove that R is an equivalence relation on Z. List the elements of the equivalence class [0]. Prove that [1] = [2]. Prove that [1] = {b Epsilon Z: b Strictly Equivalent To 1 (mod3) or 6 Strictly Equivalent To 2 (mod 3)}.

Solution

a)

a^2=a^2 mod 3 for any a in Z

Hence, aRa

SO, R is reflexive

Let, aRb so a^2=b^2 mod 3

Hence, bRa

So, R is symmetric

Let, aRb and bRc

So, a^2=b^2=c^2 mod 3

HEnce, aRc

Hence, R is transitive and an equivalence relation

b)

a^2=0 mod 3

So, 3|a^2

BUt, 3 is prime so 3|a

SO, [0]=all multiplies of 3

c)

1^2=1 mod 3

2^2=4=1 mod 3

HEnce, 1R3

HEnce, 1 is in [2] and 2 is in [1]

So, [1]=[2]

d)

LEt, a be in [1]

a^2=1 mod 3

a cannot be 0 mod 3

Case 1. a=3k+1

a=1 mod 3

a^2=1 mod 3

Case 2. a=3k=2

a=2 mod 3

a^2=4=1 mod 3

HEnce, proved