On A=ZxZ, define the relation (x,y)~(a,b) if a+y=x+b. A. Show that this is an eq

Question

On A=ZxZ, define the relation (x,y)~(a,b) if a+y=x+b.

A. Show that this is an equivalence relation.
B. Describe [(1,2)].
C. Describe the equivalence classes in general, i.e., for fixed a and b, describe [(a,b)].

Explanation / Answer

A) In order to see if this is an equivalent relation we need to show 3 things: (a,b) ~ (a,b). (Reflexivity) if (a,b) ~ (c,d) then (c,d) ~ (a,b). (Symmetry) if (a,b) ~ (c,d) and (c,d) ~ (e,f) then (a,b) ~ (e,f). (Transitivity) Reflexivity: a+b = a+b so (a,b) ~ (a,b) Symmetry: if (a,b)~(c,d) then c+b = a+d So a+d = c + b (this is a normal equal sign) => (c,d) ~ (a,b) Transitivity if (a,b) ~ (c,d) and (c,d) ~ (e,f) then we know: c+b = a+d from (a,b) ~ (c,d) c+f = d+e from (c,d) ~ (e,f) subtracting the equations we get b-f = a-e So e+b = a+f => (a,b) ~ (e,f) So the relation is an equivalence relation B) Notice that (1,2) ~ (2,3) ~ (3,4) ~ (4,5). So a good way to describe it would be [(x, x+1)]. Notice that the difference between the numbers is always 1. C) If we were to use any (a,b), we'll notice that the difference is always b-a, so [(a,b)]={(x,x+b-a) for all integers x}

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