A question about the definition of an "ordered pair"

[Dale10101]

Am trying to work myself into an understanding of the ZF Set theory.

Given:

Definition of an “ordered pair” An ordered pair (a, b) is defined to be (a, b) = { {a}, {a, b} }

Now ... in writing:

{ {dog}, {horse, donkey}}

... have used the ZF definition of an ordered pair to define the ordered pair (a,b)

a = {dog},
b = {horse, donkey}

also, extending this concept further and writing:

{ {dog}, {cat, mouse, lizard}, {horse, donkey}}

have defined the ordered triplet (a,b,c)

a = {dog},
b = {horse, donkey}, and
c = {cat, mouse, lizard}

or, is there an additional requirement that a succeeding element must contain its preceding element as in the case of the integers ... or is that a next issue in the development of the theory?


Thanks for your help.

 

[pka]

Definition of an “ordered pair” An ordered pair (a, b) is defined to be (a, b) = { {a}, {a, b} }
Now ... in writing:
{ {dog}, {horse, donkey}}
... have used the ZF definition of an ordered pair to define the ordered pair (a,b)
a = {dog},
b = {horse, donkey}

Absolutely not! The set \(\displaystyle \{ \{dog\}, \{horse, donkey\}\}\) is absolutely not an ordered pair.


An ordered pair with two different elements is a set of two non-empty sets, one of which is a singleton set that is a proper subset of the other set which has two elements.

Example: the set \(\displaystyle \{\{a,2\},\{2\}\}\) defines the ordered pair \(\displaystyle (2,a)\).

It works this way: The singleton set identifies the first ordinate of the pair and the other element in the two element set determines the second coordinate.

There is a special case for a pair like \(\displaystyle (a,a)\) and it is \(\displaystyle \{\{a\}\}\) because \(\displaystyle \{\{a\},\{a,a\}\}=\{\{a\}\}\)

 

[Dale10101]

OK, to summarize what I have learned about an “ordered pair”

1) First of all, I believe I do understand how to convert, for example {{a,2},{2}} = {{2,a},{2}} = {{2}, {a,2}} = {{2}, {2,a}} into the notational equivalent (2,a).

The idea is that you start with an unordered set containing the elements that you wish to order i.e. {a,2} and select the element that you wish to come first and make it a second unordered set. Since this new set contains but one element of the original unordered set it is automatically a proper subset of the original unordered set, i.e. {2} is a proper subset of {a,2}. Putting the two unordered sets into braces creates the unordered set {{2},{2,a}} which is by definition (2,a). Pedagogically,the key is to FIRST write the unordered set that you wish to order, and THEN select the element that you wish to be the first element of the ordered set.

One could have started with {a,2} and selected “a” as the desired first element and create {{a}, {a,2}} = (a,2). I think I have this right.

2) Second, I think I started looking at this problem from the wrong end of telescope, that is, as though ZFC preceded the common notation for an ordered pair (a,b) or ordered triplet (a,b,c) vis-à-vis, its intuitive representation as a point in 2,3 geometric space.

In that mode I was trying to imagine why ZFC choose to produce this definition and becoming confused as to what they were trying to achieve.

That was, of course, totally misguided. ZFC did not generate (a,b), it was up to ZFC to “reverse engineer” a definition for an ordered pair that conformed with the preexisting definition and notation i.e. (a.b) etc. I know, duh, but that’s what happens when you look through the wrong end of a telescope, you end up with a microscope, and getting lost in the trees without seeing the forest is virtually inevitable.

3) So, indeed, nothing profound in the definition (a,b) = {{a},{a,b}}, as per M. HallsofIvy.

I think if ZFC had preceded the Cartesian notation, the definition would still have been created not because it would have added content to mathematics but because it’s meaning is so much more intuitive and so much more efficient, less cumbersome. I mean doing algebra using the notation {{a}, {a,b}} would be like doing multiplication with Roman numerals!

4) Can I get confirmation from someone if I have this right at last? I hope so because many things now seem to make more sense.

 

[pka]

3) So, indeed, nothing profound in the definition (a,b) = {{a},{a,b}}, as per M. HallsofIvy.

I think if ZFC had preceded the Cartesian notation, the definition would still have been created not because it would have added content to mathematics but because it’s meaning is so much more intuitive and so much more efficient, less cumbersome. I mean doing algebra using the notation {{a}, {a,b}} would be like doing multiplication with Roman numerals!

4) Can I get confirmation from someone if I have this right at last? I hope so because many things now seem to make more sense.

I think that you need to study this webpage.

You can see that there are several different definitions of ordered pairs. You will also note that ZFC is not even mentioned in that write up.

 

[Dale10101]

follow up

I think that you need to study this webpage.

You can see that there are several different definitions of ordered pairs. You will also note that ZFC is not even mentioned in that write up.

These two sentences seem germane to what had heretofore been my point of confusion:

"The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property."

and

"If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set.^[1] Several set-theoretic definitions of the ordered pair are given below."

For me, the clarification is that (a.b) is defined by its characteristic property and that the notation of (a,b) stands for the same thing in set theory as it had before its definition in set theory. To me that was not obvious. It seems that sometimes the same notation is used differently in different branches of mathematics and by different mathematicians. I think I have my wires uncrossed on this point.

Also, yes, I note that ZFC is not the "end all", "be all" of set theory, there are different formulations. Thanks.

 

[HallsofIvy]

Am trying to work myself into an understanding of the ZF Set theory.

Given:

Definition of an “ordered pair” An ordered pair (a, b) is defined to be (a, b) = { {a}, {a, b} }

Now ... in writing:

{ {dog}, {horse, donkey}}

... have used the ZF definition of an ordered pair to define the ordered pair (a,b)

NO, this is not an ordered pair because the first member of {horse, donkey} is NOT "dog". That is, the "a" in {a, b} is NOT the same "a" as in {a}. If this were {{dog}, {dog, donkey}} or {{horse}, {horse, donkey}} THEN they would be "ordered pairs", the first defining the ordered pair (dog, donkey) and the second the ordered pair (horse, donkey).

a = {dog},
b = {horse, donkey}

also, extending this concept further and writing:

{ {dog}, {cat, mouse, lizard}, {horse, donkey}}

have defined the ordered triplet (a,b,c)

a = {dog},
b = {horse, donkey}, and
c = {cat, mouse, lizard}

or, is there an additional requirement that a succeeding element must contain its preceding element as in the case of the integers ... or is that a next issue in the development of the theory?


Thanks for your help.

 

[Dale10101]

Yes

NO, this is not an ordered pair because the first member of {horse, donkey} is NOT "dog". That is, the "a" in {a, b} is NOT the same "a" as in {a}. If this were {{dog}, {dog, donkey}} or {{horse}, {horse, donkey}} THEN they would be "ordered pairs", the first defining the ordered pair (dog, donkey) and the second the ordered pair (horse, donkey).

Yes, I see the point now, you start with the unordered set you wish to order, in this case {horse, donkey} and select the element you wish to place first as a first coordinate, say "donkey", place it in braces to make it a singleton, {donkey} and then place the singleton and the original unordered set in inclusive braces to produce {{donkey}, {horse, donkey} }, an ordered set. Thanks.