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Equivalence Relations and Classes

Define a relation R on N × N by (a, b)R(c, d) if and only if a + b = c + d.
(i) Prove that R is an equivalence relation on N × N.
(ii) Let S denote the set of equivalence classes of R. Show that there is a 1-1 and
onto function from S to N.

I'll attempt the first part

a + b = a + b so (a, b)R(a, b) i.e. R is reflexive
c + d = a + b so (a, b)R(c, d) = (c, d)R(a,b) i.e. R is symmetric
Unsure how to show R is transitive ...

No idea about how to derive the equivalence classes or show that there's a 1-1 and onto function from S to N


Solution Summary

Equivalence Relations and Classes are investigated.