MHB Binary Relations and Equivalence Classes | Proving R is an Equivalence Relation

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The discussion focuses on proving that the binary relation R, defined by R={(x,y)∈ R×R:cos(x)=cos(y)}, is an equivalence relation. The user has established reflexivity and is seeking help with symmetry and transitivity. It is clarified that symmetry holds since cos(x) = cos(y) implies cos(y) = cos(x). For transitivity, it is noted that if cos(x) = cos(y) and cos(y) = cos(z), then cos(x) = cos(z) also holds true. The equivalence classes are identified as consisting of values that differ by integer multiples of 2π and also include negatives, leading to the conclusion that the equivalence class [x] includes both x + 2kπ and -x for all integers k.
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So the question I am trying to solve is this:

Define a binary relation R on R as follows: R={(x,y)∈ R×R:cos⁡(x)=cos⁡(y)}
Prove that R is an equivalence relation, and determine its equivalence classes.

I've figured out the first two requirements for being a binary relation:

1. cos(x) = cos(x)
2. cos(x) = cos(x + 2kpi)

I don't know how to go about solving the third requirement for being a binary relation because there is no z to work with.
If someone could show me how to find the third requirement, that would be great.

Thank you
 
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JProgrammer said:
So the question I am trying to solve is this:

Define a binary relation R on R as follows: R={(x,y)∈ R×R:cos⁡(x)=cos⁡(y)}
Prove that R is an equivalence relation, and determine its equivalence classes.

I've figured out the first two requirements for being a binary relation:

1. cos(x) = cos(x)
2. cos(x) = cos(x + 2kpi)

I don't know how to go about solving the third requirement for being a binary relation because there is no z to work with.
If someone could show me how to find the third requirement, that would be great.

Thank you

Hi JProgrammer! Welcome to MHB! ;)

From wiki, an equivalence relation requires:

For all a, b and c in X:
1. a ~ a. (Reflexivity)
2. a ~ b if and only if b ~ a. (Symmetry)
3. if a ~ b and b ~ c then a ~ c. (Transitivity)

The equivalence class of a under ~, denoted [a], is defined as $[a]=\{b\in X\mid a\sim b\}$.​

You have shown the first, since indeed for any $x$ we have $\cos x = \cos x$.
Can you show 2 and 3, and can you find the equivalence classes?
 
You have shown the first said:
I don't understand what you mean. For the symmetric requirement, I have shown it as: cos(x) = cos(x + 2kx). As for the transitive requirement, that is what I need help with. I don't know what that would be since there is no z to work with.

Also, I don't know how to find the equivalence classes.
 
JProgrammer said:
I don't understand what you mean. For the symmetric requirement, I have shown it as: cos(x) = cos(x + 2kx). As for the transitive requirement, that is what I need help with. I don't know what that would be since there is no z to work with.

Also, I don't know how to find the equivalence classes.

Ah, but that is not symmetry.
For symmetry we need that for all $x,y$ we have that $xRy$ if and only if $yRx$.
That means that for all $x,y$ we need that $\cos x = \cos y$ if and only if $\cos y = \cos x$.
This is trivially true. It follows from how the equal sign ($=$) is defined, which is also an equivalence relation.

For transitivity we need that for all $x,y,z$ we have that if $\cos x = \cos y$ and $\cos y=\cos z$, that we also have that $\cos x = \cos z$.
Again, this is trivially true.

That leaves the equivalence classes.
You have found that $x$ is equivalent to $x+2k\pi$, since $\cos(x) = \cos(x+2k\pi)$.
So that might suggest that $[x]=\{x+2k\pi \mid k\in \mathbb Z\}$.
But how about $-x$? Don't we have that $\cos(x)=\cos(-x)$?
That would imply that $-x$ should also be in the equivalence class $[x]$. (Worried)
 
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