1. ## 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 ≅?

2. 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 ≅?

for (a), 2 ordered pairs (a,b) and (c,d) belong to the same equivalent class if $\displaystyle \frac{a}{b} = \frac{c}{d}$

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..

3. 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.

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

$\displaystyle A=\left\{\frac{1}{3},\frac{2}{4},\frac{-4}{-8},\frac{3}{9},\frac{1}{5},\frac{3}{6}\right\}$

you see that $\displaystyle \frac{1}{3} = \frac{3}{9}$.
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) $\displaystyle aRb \Longleftrightarrow a^3 \equiv b^3 {\bmod 7}$

let $\displaystyle a,b,c \in \mathbb{Z}$

i) of course, $\displaystyle a^3 \equiv a^3 {\bmod 7}$

ii) suppose aRb, i.e. $\displaystyle a^3 \equiv b^3 {\bmod 7}$. then $\displaystyle a^3-b^3 \equiv 0 \bmod 7 \Longleftrightarrow b^3-a^3 \equiv 0 \bmod 7 \Longleftrightarrow b^3 \equiv a^3 \bmod 7$ thus, bRa.

iii) suppose aRb and bRc.

so, $\displaystyle a^3 \equiv b^3 {\bmod 7}$ and $\displaystyle b^3 \equiv c^3 {\bmod 7}$

meaning,
$\displaystyle a^3 - b^3 \equiv 0 {\bmod 7}$ ----- (1)
$\displaystyle b^3 - c^3 \equiv 0 {\bmod 7}$ ----- (2)

adding (1) and (2) you get
$\displaystyle a^3 - c^3 \equiv 0 {\bmod 7} \Longleftrightarrow a^3 \equiv c^3 \bmod 7$. thus aRc.

hence, R is an equivalence relation..

for the last one.. $\displaystyle a \cong b \Longleftrightarrow a \equiv b \bmod 7$
simply, $\displaystyle a$ and $\displaystyle b$ have the same remainder when they are divided by 7.

so, the equiv. class are $\displaystyle 7\mathbb{Z} + n$ where $\displaystyle n = 0,1,...,6$