# Set Theory Proof: Proving lx-yl ≤2r for All x,y εA

• Seda
In summary, the problem states that there exists a set x such that for all x,y in A, there exists a lower bound on the distance between x and y. It is not necessarily true that for all x,y in A, this distance is less than 2r.
Seda

## Homework Statement

This is the problem stated verbatim. xo is supped to be x with a subscript o.

Suppose that A is a set and there exists xo ε A for which lx-xol ≤ r. Is it necessarily true true that for all x,y εA, we will have lx-yl ≤2r?

## Homework Equations

Well, this problem is just supposed to test our ability to formulate proofs, so I can't think of many helpful equations.

I do know lxl < r is the same thing as -r<x<r

## The Attempt at a Solution

I can't even think of how to start proving this. In desperation, I tried to think of a counterexample to prove this untrue, but none came to mind. It's also weird that r is such an arbitrary value or variable or something. Nothing is stated in the problem on what it's supposed to represent. I think it means any picked real number but I'm not sure.

Any ideas?

For clarity, x is supposed to be any element of A correct? If so, then think about using the triangle inequality if it's available.

Ok sweet, I completely forgot about the triangle inequality (yes it is available.) However, I am not sure on how to implement it exactly.

So lx+-xol≤ lxl +lxol by the triangle inequality.
lx-yl≤ lxl+lyl ditto.

SO now my equation has become this:

if lxl+lxol≤r
is it true that lxl + lyl≤2r.

Because if I prove that, the desired proof follows?

Last edited:
No, that's not true. Just because |x - xo| is less than or equal to r doesn't mean that |x| + |xo| is too. Really, think more intuitively about what |x - xo| being less than r says.

I'm lost still, mainly beacuse of the confusion between x and xo and what r represents.

Its the difference between a picked value and any other value in a set? sorry If I'm coming across as a total dimwit, but I'm not experiences any revelations here..

Prove that the value of the difference between any two members of a set is always less than twice the value always greater that the difference between any member and one specified member of the same set?

If |x-x0| < r you can think of it as saying that the disk of radius r about x0 encloses the set. What can you then conclude?

I'm an idiot too often, sorry.

So obviously this set can be represented by a circle, where each point is obvious less that a diameter's width away from every other point. diameter = 2r.

I will say that mathdope is a misleading name because you've helped me a ton. Now I just need to ship that explanation up into a mathematical set logic proof rather than a verbal diagram proof, but I thinki should be good to go. Thanks a million

Seda said:
I will say that mathdope is a misleading name because you've helped me a ton. Now I just need to ship that explanation up into a mathematical set logic proof rather than a verbal diagram proof, but I thinki should be good to go. Thanks a million

A picture should help with the proof. Draw a circle with x and y at random points and x0 at the center. Then try to fit the triangle inequality in with your picture.

Sorry to continue to be such a nag, but is there any way I could prove this without referring to the circle at all (my proffessor is crazy about not using diagrams). I understand that lx-yl represents the distance between any two points, which obviously can't exceed 2*r, but then I think I'm assuming the thing I want to prove. Its easy to explain the proof in the context of the circle idea, but without it I seem to stumble even more when I try to just tplainly use inequalities.

Wow, this is the most hopeless I've even felt, and I'm on the dean's list of students too... I really hope I'm just really tired and not normally this stupid.

Here's a tip that will get you through a good chunk of analysis problems: add 0 inside the absolute value sign in a "creative" way.

Hopefully, this will be my last post.

We have x,y as elements of A. Thus we know that

lx-xol≤r

and we know that
ly-xol≤r

based on the question.
So we can add these together to get

lx-xol+ly-xol≤2r

which is the same as

lx-xol + lxo-yl≤2r

By the triangle inequality, we know that

l(x-xo)+(xo-y)l ≤ lx-xol + lxo-yl

So we combine these to get

l(x-xo)+(xo-y)l ≤ lx-xol + lxo-yl ≤ 2r

Thus, l(x-xo)+(xo-y)l ≤ 2r
Finally, lx-yl≤ 2r

Please say that's doable, I want some sleep!
Edit: I have done the bolded step before, even though it's awkward, but my teacher says it's fine in other proofs.

Looks like you got the jist.

you shouldn't apologize for not knowing something =)

## 1. What is set theory?

Set theory is a branch of mathematics that deals with the study of collections of objects or elements, called sets. It provides a foundation for other areas of mathematics, such as algebra and calculus, and is used to develop formal proof techniques.

## 2. What is a proof in set theory?

A proof in set theory is a logical argument that demonstrates the truth of a statement or proposition about sets. It involves using the axioms and rules of set theory to show that a given statement is true.

## 3. What does lx-yl ≤2r mean?

lx-yl ≤2r is an inequality that states that the absolute difference between the elements x and y in set A is less than or equal to twice the radius r. In other words, the distance between x and y is smaller than or equal to twice the size of the radius r.

## 4. How do you prove lx-yl ≤2r for all x,y εA?

To prove lx-yl ≤2r for all x,y εA, we need to use the axioms and rules of set theory to show that the inequality holds true for any pair of elements x and y in set A. This can be done by constructing a logical argument that follows the rules of set theory and shows that the statement is true for all possible cases.

## 5. What is the importance of proving lx-yl ≤2r in set theory?

Proving lx-yl ≤2r in set theory is important because it allows us to establish a fundamental property of sets, namely that the distance between any two elements in a set is limited by the size of the radius. This proof can then be used in other mathematical contexts to build more complex arguments and proofs.

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