Thread: Equivalence classes HELP!!!

(a) Let A = {(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}. Define a relation R on A as follows: (a,b)R(c,d) if ad=bc. List the equivalence classes of R.
(b) Let a,b ∋ Z
(i) Define aRb and only if a^3 ≡ b^3 (mod 7).
Prove that R is an equivalence relation on Z.
(ii) Define a ≅ b if and only if a ≡ b (mod 7), What are the equivalence classes for ≅?

Please also explain your answers/working out as you go along (if possible). thanks

Originally Posted by Unt0t

(a) Let A = {(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}. Define a relation R on A as follows: (a,b)R(c,d) if ad=bc. List the equivalence classes of R.
(b) Let a,b ∋ Z
(i) Define aRb and only if a^3 b^3 (mod 7).
Prove that R is an equivalence relation on Z.
(ii) Define a ≅ b if and only if a ≡ b (mod 7), What are the equivalence classes for ≅?

Please also explain your answers/working out as you go along (if possible). thanks

for (a), 2 ordered pairs (a,b) and (c,d) belong to the same equivalent class if

now, how do you prove that R is an equivalence relation?
i) aRa
ii) aRb then bRa
iii) if aRb and bRc, then aRc..

so, i believe, it is better that you post what you have done already so that we can check your work..

To be quite honest i'm very lost... I understand what you're saying, but I have no idea where to start or how to go about answering the questions.

for a)
A = {(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}
let us write it in this way..



you see that .
therefore, they belong to the same equivalence class..

the other equivalence class contains (2,4), (-4,-8), and (3,6)

another equivalence class contains (1,5).


for b)

let

i) of course,

ii) suppose aRb, i.e. . then thus, bRa.

iii) suppose aRb and bRc.

so, and

meaning,
----- (1)
----- (2)

adding (1) and (2) you get
. thus aRc.

hence, R is an equivalence relation..

for the last one..
simply, and have the same remainder when they are divided by 7.

so, the equiv. class are where