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**Set Theory**

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

You have still yet to define union in terms of a binary operation. Please, answer the question. You state that union is a binary operation, all I ask is that you define it as such. I'm not looking for examples, I'm look for you to define union as a mapping from a set AxA to A.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

In your definition, how is A defined?

For it to be a binary operator, the mapping must be defined on all of AxA. You seem to be saying that X and Y are elements of A, no?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Oddly enough, I didn't think I had a point. I was really just trying to understand your definition. But perhaps it can be saved.

If we have A be the universal set, then we define the binary operation of union:

This is defined over all of AxA in the same way we can take the union of any two sets. It is also uniquely defined, as all unions are.

Only one problem. The universal set contains all sets. So the universal set must contain itself, as it is a set. But this is not allowed by the Axiom of Regularity* (under ZFC set theory). So such a universal set can't exist.

*Axiom of Regularity states:

This is saying in a general way that A can not be in A.

Proof by contradiction. Assume A is in A. **Then A is in {A} intersect A.** There must exist a b in A such that b intersect {A} = null. Since the only element of {A} is A itself, it must be that b = A. So we replace "b intersect {A} = null" with **"A intersect {A} = null"**. So A is in {A} intersect A and A intersect {A} is null. Contradiction.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Remove this after cleanup:

Something from wikipedia....

Some philosophers take a more radical approach, holding that some contradictions are true, and thus a theory's being inconsistent is not always an indication that it is incorrect. This view, known as dialetheism, is motivated by several considerations, most notably an inclination to take certain paradoxes such as the Liar and Russell's paradox at face value.

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What if you guys just make two or more different threads on different kinds of set theory?

*Last edited by John E. Franklin (2007-02-26 13:22:03)*

**igloo** **myrtilles** **fourmis**

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**Sekky****Member**- Registered: 2007-01-12
- Posts: 181

Ricky wrote:

You have still yet to define union in terms of a binary operation. Please, answer the question. You state that union is a binary operation, all I ask is that you define it as such. I'm not looking for examples, I'm look for you to define union as a mapping from a set AxA to A.

Are you serious? You want me to list every possible tuple in existance? You're clinically insane, I'd like to see you do that for an infinite set.

Here it is for my example above

Here we go

There you go, have fun writing it out for every possible set imaginable. My previous posts are the method in which it is defined, not by the mapping. Are you telling me scalar addition is defined by writing out every single possible tuple and a permutation? Of course not, it's defined by a couple of predicate rules.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Yet again, you post *an example*. What I am asking for is a general definition. For example, a general definition for addition on the natural numbers is:

Where S(n) is the successor function of n.

This defines addition for all natural numbers. I am looking for the same general definition for union. There are two ways I see that you can attempt to do so:

1. Take the universal set and create a mapping. Problem is, the universal set doesn't exist under ZFC.

2. Take a set which contains the elements of sets A and B, and create a mapping. Problem is, you can't define a set, in general, which contains the elements of A and B without taking the union of them.

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**medivijaysagar****Member**- Registered: 2008-01-09
- Posts: 1

hi ganesh, first of all i would like to thank you for this great stuff. Any guy will find it very useful in any mode of his life. i fell lucky to find this site. Thanks once again and keep going........

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,828

Thanks, medivijaysagar!

It is a combined effort of all the members of this forum.

Welcome to the forum!!!

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,828

**Cartesian Products**

1)

2)

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**DeanPemberton****Member**- From: USA
- Registered: 2014-10-30
- Posts: 20
- Website

Set theory is a mathematical theory of well determined collections, sets of objects that are called members or elements. Set theory is very simple theory but the concept of it is little bit difficult.

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**loveMath****Member**- Registered: 2015-02-18
- Posts: 3

Sorry I am new here. If possible, can anybody help me? My question is: given the two sets A={x} and B={{x}} are they equal?

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No, they are not.

Is having a carrot in a box and a carrot in a box in a box the same?

Please make a new thread in the help me section

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**loveMath****Member**- Registered: 2015-02-18
- Posts: 3

thanks I will make a new post. I just did know how.

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**Toshi****Member**- Registered: 2016-02-18
- Posts: 1

I have been trying for a few days to understand this text of Zermelo, taken from his article of 1908 "Investigations in the Foundations of Set Theory", but I just do not seem to get it. I would really appreciate any help in deciphering it.

"13. Introduction of the product. If M is a set different from 0 and a is anyone of its elements, then according to No.5 it is definite whether M = {a} or not. It is therefore always definite whether a given set consists of a single element or not.

Now let T be a set whose elements, M, N, R, . . ., are various (mutually disjoint) sets, and let S1 be any subset of its union ST. Then it is definite for every element M of T whether the intersection [M, 8 1 ] consists of a single element or not. Thus all those elements of T that have exactly one element in common with 8 1 are the elements of a certain subset T 1 of T, and it is again definite whether T 1 = T or not. All subsets S1 of ST that have exactly one element in common with each element of T then are, according to Axiom III, the elements of a set P = T, which, according to

Axioms III and IV, is a subset of union T and will be called the connection set [Verbindungsmenge] associated with T or the product of the sets M, N, R, . . .. If T = {M, N}, or T = {M, N, R}, we write T = MN, or T = MNR, respectively, for

short. "

I just do not understand why it is called "product" and how {M,N} can become MN here. Not in general therefore, but in this text. Thank you.

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