Let n be an element of the positive integers and let --- be defined on Z by r---
ID: 1942740 • Letter: L
Question
Let n be an element of the positive integers and let --- be defined on Z by r---s if and only if r-s is divisible by n, that is, if and only if r-s=nq for some q that is an element of Z.--- equivalence relation symbol
Z of course means integer
a.) Show that --- is an equivalence relation on Z. (It is called congruence modulo n )
b.) The cells of this partition of Z are residue classes modulo n in Z. Write the residue classes modulo n in Z using the notation {...,#,#,#,...} for these infinite sets.
Explanation / Answer
I don't think I can do b.
Reflexive: For all r in Z, r ~ r = r - r = 0 = nq for some q in Z
Symmetric: For all r, s in Z, if r ~ s, then s ~ r
r ~ s -> r - s = nq
s ~ r -> s - r = -1(r - s) = nq
Transitive: For all r, s, t in Z, if r ~ s and s ~ t, then r ~ t
(r - s) + (s - t) = r - t -> r ~ t
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