Let D be the relation on Z defined as follows:

for all m,n in Z m D n <==> 3|(m^2 - n^2)

1.prove that D is an equivalence relation and

2.describe the distinct equivalence classes of D.

I don't know how to go abt it(Worried)

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- December 11th 2008, 07:37 AMNidhiSequivalence classes
Let D be the relation on Z defined as follows:

for all m,n in Z m D n <==> 3|(m^2 - n^2)

1.prove that D is an equivalence relation and

2.describe the distinct equivalence classes of D.

I don't know how to go abt it(Worried) - December 11th 2008, 08:51 AMTheEmptySet
To be an equivelence relation it must be

Reflexive

Symmetric

and

Transative if

Reflexive yes becuause

symmetric yes because if

for some integer q, factoring we get

so

Transative

and

solving the 2nd for we get

sub this into the first one to get

This tells us that

for part two think graphically. Good luck. - December 11th 2008, 10:57 AMSoroban
Hello, NidhiS!

I'll do part 1 . . .

Quote:

Let be the relation on defined as follows:

. . for all

1. Prove that is an equivalence relation.

. . . . . . . . (The difference of their squares is a multiple of 3.)

Reflexive: .Is ?

. . . . . Yes!

Symmetric: . ?

. . We have: .

. . Then: . . . . Yes!

Transitive: . ?

. .

. . Add: . . . . Yes!

is reflexive, symmetric and transitive.

Therefore, is an equivalence relation.