Proofs help Let the relation R be defined on the set Z of al

Proofs help

Let the relation R be defined on the set Z of all integers by a R b if and only if a^2 - b^2 is a multiple of 5. Determine whether R is an equivalence relation. Either prove that R is an equivalence relation or specifically demonstrate which properties of an equivalence relation it fails to have. Determine [1] = {n epsilon Z | 1R n}.

Solution

1.

m^2-m^2=0 is a multiple of 5

HEnce, mRm

SO, R is reflexive

2. Let m^2-n^2 is a multiple of 5

Hence, n^2-m^2 is a multiple of 5

Hence, R is symmetric

3. .Let m^2-n^2 and n^2-k^2 be multiple of 5

Then, m^2-n^2+n^2-k^2=m^2-k^2 is multipleo f 5

Hence, R is transitive

So, R is equivalence relation.

To find equivalence classes we need to look at residues of squares modulo 5

To integers ,m,n are in teh same equivalence class if their squares give same residue modulo 5

0^2=0

(+-1)^2=1 mod 5

(+-2)^2=-1 mod 5

So three equivalence classes

[0],[1],[2]

Hence, [1]={n in Z, n=5k+1 or 5k-1 for some k in Z}