Help with an Equivalence Relation?

[katye333]

Hello all, I have an equivalence relation that I need some help with. Normally I find these to be fairly simple, however I'm not sure if I'm over-thinking this one or if it's just tricky.

For the relation: aRb $\Longleftrightarrow$ |a| = |b| on $\mathbb{R}$ determine whether it is an equivalence relation.

Reflexive: Would it really be reflexive? If a = -2, then wouldn't |a| = +2?

Or would it be reflexive, since all a's are contained in a?

 

[Fernando Revilla]

Reflexive: Would it really be reflexive? If a = -2, then wouldn't |a| = +2?
Or would it be reflexive, since all a's are contained in a?

A relation $R$ is reflexive on a set $S$ if and only if $aRa$ for all $a\in S$. In our case $|a|=|a|$ for all $a\in \mathbb{R}$ i.e. $aRa$ for all $a\in \mathbb{R}$, which implies that $R$ is reflexive on $\mathbb{R}$.

 

[Evgeny.Makarov]

You can prove a general theorem. Let $A$, $B$ be sets and $f:A\to B$ a total function. Let a relation $R$ on $A$ be defined by $a_1Ra_2\iff f(a_1)=f(a_2)$. Then $R$ is an equivalence relation. In this case $f:\mathbb{R}\to\mathbb{R}$ is the absolute value function. You can also apply this theorem to $f(x)=\mathop{\text{sgn}}(x)=\begin{cases}1&x>0\\ 0&x=0\\ -1&x<0\end{cases}$, $f(x)=\lfloor x\rfloor$, etc.

Note also that if we take not a function $f$, but an arbitrary relation $F\subseteq A\times B$ and define \[a_1Ra_2\iff \text{there exists a }b\in B\text{ such that }a_1Fb\text{ and }a_2Fb\] then the statement does not hold in general. (Which property of an equivalence relation gets violated?) For example, it does not hold if $A=B$ is a set of people and $aFb\iff b$ is a friend $a$.

 

[katye333]

Thank you both for the responses.
I don't know why that one confused me, while none of the others did. :p

 

[blue_raver22]

Hello,

Thank you for reaching out for help with your equivalence relation. Let's first define what an equivalence relation is. An equivalence relation is a relation that is reflexive, symmetric, and transitive. This means that for the relation aRb, it must satisfy the following properties:

1. Reflexive: For all elements a, aRa must be true.
2. Symmetric: If aRb is true, then bRa must also be true.
3. Transitive: If aRb and bRc are true, then aRc must also be true.

Now, let's apply these properties to the relation you have provided, aRb $\Longleftrightarrow$ |a| = |b| on $\mathbb{R}$.

1. Reflexive: To determine if a relation is reflexive, we must check if aRa is true for all elements a. In this case, it means we must check if |a| = |a| for all real numbers a. And the answer to that is yes, because the absolute value of any number is always equal to itself. So, the relation is reflexive.
2. Symmetric: To check if a relation is symmetric, we must check if whenever aRb is true, bRa is also true. In this case, it means we must check if |a| = |b| implies |b| = |a|. And the answer to that is also yes, because the absolute value of a number is independent of its sign. So, the relation is symmetric.
3. Transitive: To check if a relation is transitive, we must check if whenever aRb and bRc are true, aRc is also true. In this case, it means we must check if |a| = |b| and |b| = |c| implies |a| = |c|. And the answer to that is yes, because if two numbers have the same absolute value, then their absolute values must also be equal. So, the relation is transitive.

Therefore, the relation aRb $\Longleftrightarrow$ |a| = |b| on $\mathbb{R}$ is an equivalence relation. I hope this helps clarify any confusion you may have had. If you have any further questions or need any more help, please don't hesitate to ask.

Best regards,