What Are Set Relations and Their Notations in Homework Problems?

[MostlyHarmless]

Homework Statement

I've actually got a couple questions, I'll provide an example for each question, but I'm not really looking for an answer to the example, but an explanation of the concept. I have very little to go on from class notes. We've had some inclement weather in these parts leading to the campus being closed since Thursday, so we've missed two class meetings where I could have asked questions. And I'm assuming the test over this stuff is tomorrow.(Its all a mess)

First, is just basic basic Set Relation principles.

Let $A=${$1,0$}. Determine all of the Relations on A.

I just need to know what this question wants. What does it mean by Relations ON A?

Under the same section:

Let A be a nonempty set and B be a subset of ρ(A) (I think this is the power set notation). Define a relation R from A to B by xRY if x ${\epsilon}$ Y. Give an example of 2 sets A and B that illustrate this. What is R for these two sets.

Again, I'm not entirely sure what the question is asking. And I'm not entirely clear what "x R Y" is supposed to mean.

So really just clarity on notation would help tremendously.

I understand that notation conventions, particularly in this subject aren't always the same, so if there is any confusion on that I'll do my best to clear it up.

Homework Equations

The Attempt at a Solution

 

[Office_Shredder]

"Relation on A" means a relation from A to A. If you understand the definition of a relation you should be able to write all relations from A to A down fairly easily.

xRY is read "x is related to Y" and means (x,Y) is contained in R. It's the standard notation used to state when two elements are related through a relation R. Remember that x is in A, so in particular is an element of A, and Y is in B, so is a subset of A.

 

[MostlyHarmless]

So then, would it be all possible subsets of AxA?

Still a bit unclear on the second question. You said, Y is in B. Does that come from xRY? And then what is the significance of Y being a set, and x being an element of that set?

 

[Office_Shredder]

So then, would it be all possible subsets of AxA?

Yes.

Still a bit unclear on the second question. You said, Y is in B. Does that come from xRY? And then what is the significance of Y being a set, and x being an element of that set?

Yes, R is a relation from A to B. This means that it is a subset of A\times B. xRY is just another way of writing (x,Y) \in R. But the first entry of each pair in R is contained in A, and the second entry of each pair of R is contained in B. That's why x\in A and Y \in B

The significance only comes from the fact that it says "xRY if x \in Y". So Y better be an object that can contain x for this to make any sense!

 

[MostlyHarmless]

Thank you for your help. I haven't abandoned this. This test was just postponed so it has taken a back seat for the moment.