Question

a) For m, n e N define m nifm n is a multiple of 3. Show that - is an equivalence on N

Use Discrete Math to solve this question.

Answer 1

m ~ n
if (m^2-n^2) is a multiple of 3

1.Reflexive

   m = n
   m^2 - n^2 = 0 it is a multiple of 3

It is reflexive

2. Symmetric

   m ~ n -> n ~ m

   It is true beacuse m^2 - n^2 and n^2 - n^2 will be same the
   only difference will be in sign but otherwise the mabnitude will
   be same. So if m^2 - n^2 is a multiple of 3 so n^2 - m^2 will also
   be the multiple of 3

   It is symmetric

3. Transitive:

   m ~ n m^2-n^2 is a multiple of 3

   n ~ p n^2 - p^2 is a multiple of 3

   Then
  
   we need to check if m ~ p holds

    m^2 - p^2 - n^2 + n^2

    m^2 - n^2 + n^2 - p^2

    3*k1 + 3.*k2       as m ~ n and n~p
    3 *(k1 + k2)

    Hence m^2 - p^2 is a multiple of 3

Hence ~ is a equivalence relation