# Thread: equivalence classes

1. ## equivalence classes

I am having trouble starting out these classes, and i'm not understanding how to find a class. Any ideas on where to start would be great.

Consider the relation I on real numbers defined by xIy <=> x-y is an integer. Describe the equivalence classes of 1/2, 2, pie and 3/2

where <=> means if and only if.

What I'm doing is as follows:

[1/2]={xE real numbers : xR0} then i don't know what to do?

2. Originally Posted by scottie.mcdonald
I am having trouble starting out these classes, and i'm not understanding how to find a class. Any ideas on where to start would be great.

Consider the relation I on real numbers defined by xIy <=> x-y is an integer. Describe the equivalence classes of 1/2, 2, pie and 3/2

where <=> means if and only if.

What I'm doing is as follows:

[1/2]={xE real numbers : xR0} then i don't know what to do?
um, no. $\left[ \frac 12 \right] = \{ x \in \mathbb{R} \mid (1/2 - x) \in \mathbb{Z} \}$

so it is the set of all real $x$ that make $\frac {1 - 2x}2$ an integer. meaning, we need $1 - 2x$ to be even.

3. how did you know to use (1/2 - x)E of integers?

4. Originally Posted by scottie.mcdonald
how did you know to use (1/2 - x)E of integers?
i am not sure what you mean. by definition that is how our relation is set up. if the difference is an integer. look back at how you relation is defined. xIy <=> x - y is an integer

5. urg, yes I see what you mean. and since 1/2-x is even by definition, that's why we have to show it's even. Thank you for the help =)

6. Here is yet another way to look at the classes.
$\left[ {\frac{1}{2}} \right] = \left\{ {\frac{1}{2} + n:n \in \mathbb{Z}} \right\}$
In general: $\left( {\forall x \in \mathbb{R}} \right)\left[ {\left[ x \right] = \left\{ {x + n:n \in \mathbb{Z}} \right\}} \right]$