Look at this web page..
You can always to some research on the web.
For this question, I would use if and only if have the same remainder when divided by .
So .
Let R be a relation on , where mod 8
I have an answer saying this is an equivalence relationship; however, I can't seem to derive it myself. Maybe it is because my understanding of what mod 8 means... the way I understand the mod 8 term is that x = y with a remainder of 8 (or multiple of 8?), but why has been used instead of ? Does it make a difference?
To check if the relation is reflexive would I say x = x + 8k for some integer k? It seems like that would not be true or only for k = 0. I'm am stuck for symmetry as well.
I think I can see transitive... If
and
then
So there is a remainder that is a multiple of 8.
Look at this web page..
You can always to some research on the web.
For this question, I would use if and only if have the same remainder when divided by .
So .
Furthermore, we conclude that R is symmetric, since
xRy <=> x = y (mod8) <=> y - x = 8k for some integer k <=> x - y = 8(-k), where (-k) is also/still an integer <=> yRx
Transitivity is simple too. I'll let you fill in the middle, after I write the hypothesis and conclusion:
To prove (xRy and yRz) => (xRz)
(xRy and yRz) <=> (y - x = 8j and z - y = 8k for integers j, k) => ....... <=> xRz