# Equivalence classes (Discrete Math)

• Dec 4th 2007, 09:19 AM
DonAvery86
Equivalence classes (Discrete Math)
Here's the original problem:
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Let R be the relation congruence modulo 4. which of the following equivalence classes are equal?
[7], [-3],[5],[-27],[32],[-14],[243],[9]
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I first began the problem by setting up four classes: 0, 1, 2, and 3. I then divided each number, for example 7 divided by 4 which yielded a remainder 3. So, 7 would be in class 3. I know how to do the positive numbers, but how would I go about doing the negative?
• Dec 4th 2007, 09:39 AM
Soroban
Hello, Don!

Quote:

Let R be the relation congruence modulo 4.
Which of the following equivalence classes are equal?
[7], [-3], [5],[-27], [32], [-14], [243], [9]

I first began the problem by setting up four classes: 0, 1, 2, and 3. . . . . Good!
I then divided each number, for example 7 divided by 4 which yielded a remainder 3.
So, 7 would be in class 3. . . . . Right!

I know how to do the positive numbers, but how would I go about doing the negative?

Use the same procedure.
If you get a negative remainder, use the modulo to "make it positive".

For example: .$\displaystyle (-27) \div 4 \:=\:-6,\;{\color{blue}\text{rem.}-3}$

. . And we know that: .$\displaystyle -3 \:\equiv\: 1 \pmod{4}$

. . Therefore: .$\displaystyle [-27] \:=\:[1]$

• Dec 4th 2007, 10:01 AM
DonAvery86
Thanks for the help. Would it be necessary for me to draw a binary relation or would it not matter?