Identities?

[bahen]

This is likely a confused query, so any and all help is welcome.

If A = x and B = x,
it does not follow that A is the same as B.

Correct? And this is another way of saying that if x is a member of set A and also of set B, it does not follow that A and B are the same set.

I would like to read more about these relations. Any direction in this regard would be appreciated.

 

[HallsofIvy]

This is likely a confused query, so any and all help is welcome.

If A = x and B = x,
it does not follow that A is the same as B.

Correct? And this is another way of saying that if x is a member of set A and also of set B, it does not follow that A and B are the same set.

I would like to read more about these relations. Any direction in this regard would be appreciated.

??? One of the conditions for an "equivalence" relation, of which "=" is the model, is that it be "transitive". That is, if A= x and B= x are true then A= B must be true.

No, that is NOT the same as saying "if x is a member of set A and also of set B, it does not follow that A and B are the same set." "Belonging to a set" is NOT an equivalence relation and should never be represented by "="! It is true that if \(\displaystyle x\in A\) and \(\displaystyle x \in B\) then it is not necessarily true that A= B. That is NOT what you wrote.

 

[Ishuda]

This is likely a confused query, so any and all help is welcome.

If A = x and B = x,
it does not follow that A is the same as B.

Correct? And this is another way of saying that if x is a member of set A and also of set B, it does not follow that A and B are the same set.

I would like to read more about these relations. Any direction in this regard would be appreciated.

As HallsofIvy said (implied), you are using the equal sign in a very non-conventional way in your statement. Assuming you meant the statement as he wrote it [x \(\displaystyle \epsilon\) A, etc.], what you are talking about is set theory. If you would like a formal introduction, you might try
https://archive.org/details/A_C_WalczakTypke___Axiomatic_Set_Theory
or for a list of free e-books on the subject
https://archive.org/search.php?query=set theory
and a fairly nice read (IMO)
http://en.wikipedia.org/wiki/Set_theory

 

[Steven G]

This is likely a confused query, so any and all help is welcome.

If A = x and B = x,
it does not follow that A is the same as B.

Correct? And this is another way of saying that if x is a member of set A and also of set B, it does not follow that A and B are the same set.

I would like to read more about these relations. Any direction in this regard would be appreciated.

A=x implies A-x=0.
Now B=x implies x-B=0. If we add the equations A-x =0 and x-B=0, then A-B = 0 or A= B. Yep, the transitive law holds.

I have a feeling that you are not telling us the whole problem.

For the record, if x is in the set A and x is in the set B then maybe A=B.

 

[stapel]

A=x implies A-x=0.
Now B=x implies x-B=0. If we add the equations A-x =0 and x-B=0, then A-B = 0 or A= B. Yep, the transitive law holds.

I have a feeling that you are not telling us the whole problem.

If you re-read the original post, you'll see that the poster refers later to element-hood. It appears that the poster is using the "equals" sign when the "is an element of" symbol is what was meant. The earlier replies to the poster explain the difference.

For the record, if x is in the set A and x is in the set B then maybe A=B.

...or maybe A is not equal to B. What might or might not be true, depending upon other circumstances, was not the question. The original poster's message asked if it would be correct to conclude that it would not logically follow (that is, it would not absolutely always to true) that A equalled B. Under the assumption stated in the earlier replies, the poster's logical statement was correct.

Apologies for the confusion. Kindly please start a new thread if you have further questions on this topic. Thank you!