# Equivalence class of 0 for the relation a ~ b iff 2a+3b is divisible by 5

jeszo

## Homework Statement

~ is a equivalence relation on integers defined as:
a~b if and only if 2a+3b is divisible by 5

What is the equivalence class of 0

## The Attempt at a Solution

 = {0, 5n} n is an integer

My reasoning for choosing 0 is that if a=0 and b=0, the relation is satisfied since 2(0)+3(0) = 0 and 5 divides 0, so 0~0
My reasoning for choosing 5n is that if a=0 and b=5n, or the other way around, then 2(0)+3(5n)=3(5n), which is a multiple of 5, and thus divisible by 5, satisfying the relation, making 0~5n

I got 0 marks for the question, but I don't know if it's because the statement isn't assembled properly, or if it's because I don't know what an equivalence class is. Please help me see my error

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## Homework Statement

~ is a equivalence relation on integers defined as:
a~b if and only if 2a+3b is divisible by 5

What is the equivalence class of 0

## The Attempt at a Solution

 = {0, 5n} n is an integer

My reasoning for choosing 0 is that if a=0 and b=0, the relation is satisfied since 2(0)+3(0) = 0 and 5 divides 0, so 0~0
My reasoning for choosing 5n is that if a=0 and b=5n, or the other way around, then 2(0)+3(5n)=3(5n), which is a multiple of 5, and thus divisible by 5, satisfying the relation, making 0~5n

I got 0 marks for the question, but I don't know if it's because the statement isn't assembled properly, or if it's because I don't know what an equivalence class is. Please help me see my error

## The Attempt at a Solution

In order to have an equivalence relation, we need both a ~ b and b ~ a, so we need both 2a + 3b and 2b + 3a to be divisible by 5.

RGV

jeszo
In response to Ray Vickson:
With 0 and 5n as the equivalence class for 0, wouldn't it still hold true that 0~0,5n~0 and 0~5n? Since, 2(5n)+3(0)=5(2n) and 2(0)+3(5n)=5(3n)?

Staff Emeritus
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It's redundant to include the zero in {0, 5n}, since if n=0, then 5n=0.

A better notation would be,  = {5n| n is an integer.}

Regarding Ray Vickson's comment, I agree with you.

The equivalence class of 0, is the set of all integers related to 0. I.e. it's the set of all integers, m, such the m~0 .