# a math problem~><

suppose that additon and multiplication have already been defined for the set of natural number N.

1. we define a relation~on N*N as follows: (p,q)~(p',q') <=>p+q'=p'+q show that defines an equivalence relation on N*N

2. it them makes sense to define equivalence classes (p,q)~: (p,q)~:={ (p',q') E N*N I(p',q')~(p,q) } the set of all these equivalence classes is denoted by (N*N)~. find all elements in the equivalence class (2,5)~. what do all these pairs of natural numbers have in common? find all elements in the equivalence class (4,2)~. what do all these pairs of natural numbers have in common?

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1

(i) p + q = p + q => (p,q) ~ (p,q)

(ii) If (p,q) ~ (p',q')

p + q' = p' + q

p' + q = p + q'

So, (p',q') ~ (p,q)

(iii) If (p,q) ~ (s,t) and (s,t) ~ (u,v)

Then p + t = s + q ; s + v = u + t

p + t + s + v = s + q + u + t

p + v = u + q

So, (p,q) ~ (u,v)

By (i) - (iii), ~ is and equivalence relation on N*N

2 If (2,5) ~ (a,b)

Then 2 + b = a + 5 where a,b in N

b = a + 3

So, the elements in the equivalence class (2,5)~ have the form (a,a + 3)

In common, the difference between the second coordinate and the first coordinate is 3

Similarly, the elements in the equivalence class (4,2)~ have the form (a,a - 2)

In common, the difference between the second coordinate and the first coordinate is -2