# Thread: Equivalence Classes and Partial Orders

1. ## Equivalence Classes and Partial Orders

I need help with the following:

f(x) = x^2 +2x + 5 and let E = {-5,-4,-3,-2,-1,0,1,2,3,4,5}

Define the relation R in E as follows: xRy if f(x) = f(y).

I need to find the equivalence classes of R and I'm completely confused, all the examples we've done in class were a lot more simple so I can't get anything out of my notes.

The second problem I'm pretty much done with:

For the following 2 relations on the set L of all living people I'm supposed to state if it is reflexive, symm, anti-symm, and transitive. Then if its a partial order indicate if the set has a maximum, minimum, maximals and minimals and explain what they mean. I'm just having trouble with the latter part.

the relations are:

xRy if X and Y have the same first name
xRy if x and y have the same major

Really appreciate help from anyone. Thanks

2. Here are hints on the first one.
$\displaystyle \begin{array}{*{20}c} x &\vline & { - 5} & { - 4} & { - 3} & { - 2} & { - 1} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline {f(x)} &\vline & {20} & {13} & 8 & 5 & 4 & 5 & 8 & {13} & {20} & {29} & {40} \\ \end{array}$
You can see that $\displaystyle \{-5,3\}$ is one equivalnce class because $\displaystyle f(-5)=f(3)$.
But $\displaystyle \{4\}$ is also a single class.WHY?

3. Originally Posted by Plato
Here are hints on the first one.
$\displaystyle \begin{array}{*{20}c} x &\vline & { - 5} & { - 4} & { - 3} & { - 2} & { - 1} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline {f(x)} &\vline & {20} & {13} & 8 & 5 & 4 & 5 & 8 & {13} & {20} & {29} & {40} \\ \end{array}$
You can see that $\displaystyle \{-5,3\}$ is one equivalnce class because $\displaystyle f(-5)=f(3)$.
But $\displaystyle \{4\}$ is also a single class.WHY?
So the eq classes are {-5,3}, {-4,2}, {-3,1},{-2,0}. As for {-1}, {4}, and {5}, are they also eq. classes? Despite that the other value isn't in the set?

Thanks

4. Originally Posted by plato
but $\displaystyle \{4\}$ is also a single class.why?
$\displaystyle 4R4$

5. Originally Posted by CrShNbRn
As for {-1}, {4}, and {5}, are they also eq. classes? Despite that the other value isn't in the set?

Thanks
Every member of the set E must belong to an equivalence class.