Understanding the Difference Between 1 and 2 in Natural Numbers

[HashTab]

first of all, id like to thank the admin for taking down a rather embarrassing post I had put up earlier. I'll try to do better this time.

N = any natural number
1 = xUx
2 = xUxUx

What exactly is the difference between 1 and 2? Is each instance of x different? If so, how?

 

[HashTab]

Someone just explained to me that sets that are not each other are different than each other because sets are different than what they are not

True or no?

 

[HOI]

Someone just explained to me that sets that are not each other are different than each other because sets are different than what they are not

True or no?

?
ANYTHING, whether a set or not, is "different than what they are not"! That's pretty much what "different" means.

I suspect that what they were trying to tell you is that a set is "defined" by what it contains. Two sets are "equal" if and only if they contain exactly the same things.

 

[HOI]

first of all, id like to thank the admin for taking down a rather embarrassing post I had put up earlier. I'll try to do better this time.

N = any natural number
1 = xUx
2 = xUxUx

What exactly is the difference between 1 and 2? Is each instance of x different? If so, how?

You wrote this- what do YOU intend "x" to mean? Normally, unless something is said to the contrary, one symbol corresponds to one mathematical object. In particular, if x represents one set then both xUx and xUxUx are the same, x, because the union of any set with itself is just itself again. xUx= x so xUxUx= (xUx)Ux= xUx= x.

However, if it is clearly stated that "x" is a "place holder", that "x" is not a specific set but just represents any set, then xUx might mean "the union of any two sets" and xUxUx might mean "the union of any three sets. (Though I would consider that poor notation.)

(What does "N is a natural number" have to do with this?)