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