Basic Topology- when doesn't the reflexive relation hold?

[Roni1985]

Homework Statement

When doesn't the reflexive relation hold?
In order for aRb to be true, aRa must hold and the other two conditions.

Homework Equations

The Attempt at a Solution

I am new to topology and am not really taking a course in topology. To me it looks like a is always equivalent to a. when is it not true?

Thanks.

 

[micromass]

Hi Roni1985!

Can you provide some more details to the question? What is R? What does this have to do with topology?

By the way, I can easily find a relation that is not reflexive, just define R to be the relation such that two elements are never in relationship to each other. Thus, the relation such that aRb is not true for all a and b.

 

[Roni1985]

Hi Roni1985!

Can you provide some more details to the question? What is R? What does this have to do with topology?

By the way, I can easily find a relation that is not reflexive, just define R to be the relation such that two elements are never in relationship to each other. Thus, the relation such that aRb is not true for all a and b.

Hi,
I am sorry,
I meant A~A and I am reading a mathematical analysis book and this topic appears under the Basic Topology chapter.

Thanks.

 

[micromass]

Hi,
I am sorry,
I meant A~A

What is R? What is ~? What is A?

Alternatively, what is the book you're reading and which problem are you doing? Maybe I can find the book...

 

[Roni1985]

What is R? What is ~? What is A?

Alternatively, what is the book you're reading and which problem are you doing? Maybe I can find the book...

lol, I am sorry, I never took topology and the notations are kind of foreign to me.

Okay, forget 'R'.
'~' = equivalent
A is a set and B is a set.

I am given a definition, "if there exists a 1-1 mapping of A onto B, A~B".

Now, when this is true, the following properties must be satisfied:

It is reflexive: A~A
It is symmetric: if A~ B, then B~A
It is transitive: if A~B and B~C, then A~C

I don't really understand the first property.
And the book is "Principles of Mathematical Analysis" by Rudin page 25.

Thanks.

 

[micromass]

Well, A~A just means here that there is a 1-1-mapping from A to A. For example,

$$A\rightarrow A:x\rightarrow x$$

is such a mapping.

It is not for every relation true that aRa, but it is true for this relation!

 

[Roni1985]

Well, A~A just means here that there is a 1-1-mapping from A to A. For example,

$$A\rightarrow A:x\rightarrow x$$

is such a mapping.

It is not for every relation true that aRa, but it is true for this relation!

When can it not be true? if I understand it correctly, there is always a 1-1 mapping from A to A.

Thanks.

 

[micromass]

Well, for the relation

$$A\sim B~\Leftrightarrow~\text{there is a 1-1 correspondence from A to B}$$

this is always true. But for other relations ~, this might not be always true.

For example, if I would define the silly relation

$$A\sim B~\Leftrightarrow~\text{there is no 1-1 correspondence from A to B}$$

then A~A is not true.

 

[Roni1985]

Well, for the relation

$$A\sim B~\Leftrightarrow~\text{there is a 1-1 correspondence from A to B}$$

this is always true. But for other relations ~, this might not be always true.

For example, if I would define the silly relation

$$A\sim B~\Leftrightarrow~\text{there is no 1-1 correspondence from A to B}$$

then A~A is not true.

Oh, I see it now.
Thanks very much for the explanation.