# Talk:Cartesian product

WikiProject Mathematics (Rated C-class, Mid-importance)
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Field: Foundations, logic, and set theory

## Untitled

I feel a link to the coproduct ${\displaystyle \coprod }$ is needed. MFH 22:20, 9 Mar 2005 (UTC)

## Cartesian product of the empty set

The opening definition of this article is surely incorrect (or at least incomplete): "Specifically, the Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y"

As that does not apply with the empty set 81.178.193.188 (talk) 21:26, 25 October 2010 (UTC)

nothing is wrong with the empty set here, the definition is precisely correct Halflingr (talk) 21:53, 15 December 2010 (UTC)

## Questions to be clarified

could u explain why the cartesian product of 2 sets is an inefficient operation to perform

I think your question is not well posed. As it stands, the answer might be: Because the sets could be uncountable sets. MFH 22:11, 9 Mar 2005 (UTC)

## deck of cards example

I must say, the deck of cards example is excellent. It caused me to link to this page instead of the very abstract one on worlfram mathworld. please keep it. --User:Brendan642

I second that. In fact, I came here to make just that comment! -- uFu 15:44, 15 July 2005 (UTC)
I also agree, and have come to the discussion page explicitly for this reason. It seems that these kind of concrete examples are extremely useful in understanding abstract mathematical concepts, and perhaps Wikipedia could use more of them in general. -- User:61.9.204.168 17/7/06 10:00 am (EST), from another computer
The deck of cards is brilliant. Does anyone know where it came from? If it's from a book I need to buy that book. 71.206.49.151 20:33, 13 November 2006 (UTC)
I'd just like to add: shouldn't the deck of cards' cartesian product actually look like this?
if A = {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2}, and
B = {♠, ♥, ♦, ♣}
then A X B = {(Ace, ♠), (Ace, ♥), (Ace, ♦), (Ace, ♣), ..., (10, ♠), (10, ♥), (10,♦), (10, ♣), ..., (2, ♠), (2, ♥), (2,♦),(2, ♣),}
/it's just in a different order shown on the other page. I think if A and B were reversed in the description, the solution would be correct.

Wity90 (talk) 21:04, 5 October 2011 (UTC) Yes that is correct, it should by corrected ASAP!!!

## delete calendar format?

Does anyone see any use in this: "The Cartesian product can be introduced by the familiar calendar format"? If not then I'll delete it. I think it confuses more than it helps. — Sebastian (talk) 00:28, July 13, 2005 (UTC)

I agree. Paul August 01:44, July 13, 2005 (UTC)

## nice article, some sugestion

What abour the properties for cartesian product? commutative, asociative,..? I like the article.

## Capitalisation

I was under the impression that "cartesian product", like "abelian group", had lost its capital letter by now. We're inconsistent about it in the article. I suggest adopting the non-capitalised spelling, just because it's the one I'm more used to.

RandomP 18:47, 29 April 2006 (UTC)

I agree, though I think there should be a brief note to the effect that both conventions are commonly used. At any rate, it should definitely be consistent one way or another. dbtfztalk 18:51, 29 April 2006 (UTC)

## Index in the projection map section

I already made a correction in the past and it was reverted back! It's not a matter of greater clarity. It's actually correct this way, and absolutely wrong the other! the ${\displaystyle i}$ index in the formula is generic and spans over all the sets in the collection, whereas the ${\displaystyle i_{0}}$ is a specific index belonging to the set ${\displaystyle I}$ of all indices, and relative to the particular projection we are performing. If you don't distinguish them, the whole thing is just wrong, that's it. Think about it. I'm pretty sure of what I'm saying. Work it out with a specific simple example to verify it, if you don't trust me.

—Preceding unsigned comment added by Roccorossi (talkcontribs)

Er, no, the definition is perfectly okay as it stands. It's okay to use definitions with nested scopes.

The main reason I reverted was that you also changed a perfectly good edit (a mere comma, but there's no reason it should go away), and that you missed at least one i. I'm also unhappy with the double subscript, because it has accessibility issues and doesn't produce the right HTML.

Can we use j instead, or ${\displaystyle i'}$, or something?

RandomP 17:01, 5 June 2006 (UTC)

I'm sorry but I have to disagree with you again. It really does matter because they refer to two different things. The "i" is the running index, so to speak, whereas the "i0" refers to the specific "dimension" onto which the projection is being performed, so they truly are different and it does matter! I went to Planet Math to check this, and I apparently am right. But they intelligently decided to use i and j instead of i an i0, so this time I followed suit and did the same thing, although I switched them because it was easier to edit the article that way.

Go here:

http://planetmath.org/encyclopedia/GeneralizedCartesianProduct.html

Please let's discuss this more instead of reverting back and forth. The last thing I want to do is give the impression of being arrogant or something, but I really am SURE of this. Let's talk about it, ok?

Roccorossi, is correct, the former was bad notation, the current is much better.

## a.k.a. cartprod

A quick Google confirms that the hypocoristic cartprod is quite widely used. I don't know if that's encyclopedic. William Avery 14:24, 27 July 2006 (UTC)

## About the example in the intro

I came to this page wondering what a Cartesian product was. The first sentence, first paragraph and ensuing formulas confused me. I didn't need to calculate it, just get an idea of what it was. I asked a friend of mine, and he explained it in the example I gave. I understood the idea immediately. Now I suck at maths. I'll grant that, but Wikipedia is supposed to be for people who suck at maths, and those who are good at maths.

The example was removed by a guy who said that it was the same example further down. When I looked I couldn't find it. This is probably because I don't understand maths language. I'd like to (humbly) request that the example I added be left in the introduction for those of us who aren't very good at maths and for who formulas make reading difficult. - Francis Tyers · 08:15, 14 May 2007 (UTC)

I merged the example into the intro text, hopefully that will be better. - Francis Tyers · 08:35, 14 May 2007 (UTC)

## Empty set

What's the Cartesian product of the empty set with itself and with a non-empty set?

I figure it must be the empty set, but I don't know. Someone put it up.

For any E E X Ø = Ø X E = Ø ; in particular Ø X Ø = Ø

Proof : forall z of E X F there's a x there's a y such that z=(x,y) and x is element of E and y is element of F -> if there's a z element of E X F neither E or F is empty. CQFD Michel42 16:28, 20 July 2007 (UTC)

I don't have a proof, but thought it was noting that a set crossed with a set containing the empty set is NOT equal to the empty set. That is to say:
${\displaystyle A\times \left\{\emptyset \right\}\neq \emptyset }$ where ${\displaystyle A\neq \emptyset }$
ialsoagree (talk) 20:57, 11 February 2010 (UTC)

## Cartesian "explosion"

Is there a technical term for the notion that a Cartesian product of several "small" variables grows quickly to an intractable level? 70.251.148.89 (talk) 02:18, 2 January 2009 (UTC)

Explosion is slang, but is generally used. If you want to be strictly formal, you could say "asymptotically intractable".195.235.92.26 (talk) 08:46, 28 August 2009 (UTC)

## Functions/ordered tuples

I'm not a set theorist or anything but lemme get one thing straight:

Functions are defined as ordered triplets (X,Y,G). Now G is the cartesian product of X and Y, which needs the concept of ordered pairs to be defined.

Order tuples on the other hand are sometimes defined as functions from the naturals to a set. Now I know they need to be functions when talking about arbitrary infinite index sets, but IS THIS DEFINITION NOT CIRCULAR?

So does it work like this: ordered pairs and ordered triples are defined in terms of sets, then we can go on and on about functions and higher than 3 ordered tuples?Standard Oil (talk) 07:30, 2 July 2009 (UTC)

Yes, that is one way to handle it. — Carl (CBM · talk) 12:03, 2 July 2009 (UTC)

Could someone please help me to proove the following formula: cartesian product of unions isn't equal to union of cartesian product: (AUB)x(CUD)=/=(AxB)U(CxD). My question is allso, which is the subset of which. Thank you for your answers! —Preceding unsigned comment added by 193.2.111.106 (talk) 12:47, 30 July 2009 (UTC)

It's not supposed to... Imagine [0,1] with itself, and [3,4] with itself. You can see the union of these two cartesian products is a proper subset of the product of their union. So generally (AxB)U(CxD) is a proper subset of (AUB)x(CUD) Standard Oil (talk) 03:05, 5 August 2009 (UTC)

## Problem?

Cartesian product seems to have circular definition with direct product at the moment (idem per idem). —Preceding unsigned comment added by 79.191.144.99 (talk) 23:19, 23 August 2009 (UTC)

## Inverse?

This may be a stupid question but I am new to this so please bare with me.

Is there an inverse to Cartesian products, like separating a set, for example the cards, back into ranks and suits?

I understand it is simpler for examples like the above and probably doesn't work as well with numbers and probably not at all with infinite sets but is there any sort of concept for this? — Preceding unsigned comment added by 109.148.177.203 (talk) 20:43, 12 June 2012 (UTC)

## product of 3 sets

Since the product of 3 sets is considered further on, I suggest that it be illustrated in the same way that the 2d Cartesian product is illustrated, for the sets A= {x,y,z} B={1,2,3} and C={Q,R,S}. This 3d diagram could illustrate the 1st several members of AxBxC as (x,1,Q), (x,1,R), which would show visually that Ax(BxC)=(AxB)xC=AxBxC. — Preceding unsigned comment added by Vlittlewolf (talkcontribs) 16:06, 23 February 2014 (UTC)

## Cartesian square definition is contradictory

It seems a bit duplicate to expose the cartesian square twice, first in the intro and, then, in the powers section. Furhtermore, intro admits the square to be a product of heterogenous sets, whereas the powers imply that the square is the product of set with itself. --Javalenok (talk) 15:21, 19 August 2014 (UTC)

## Associative Cartesian product?

David Eppstein (→‎Non-commutativity and non-associativity: I can't find any scholarly study of associative ordered pairs and set-theoretic associative Cartesian products (e.g. not in Scott/McCarty "Reconsidering Ordered Pairs"))

I couldn't find a source either. However, I vaguely remember that strings (of length n) are sometimes defined as (n-)tuples. In that context, someone might have identified string concatenation with string pairing for simplicity, thus considering the latter as associative. In this setting, the Cartesian product is associative, too, and therefore can then be used to implement the concatenation operator in regular expressions. - Jochen Burghardt (talk) 15:37, 22 October 2014 (UTC)

There are definitely associative Cartesian products out there — that exact phrase occurs in the literature. Commutative too. But I think they are for things like finite graphs rather than (what the section containing this subsection is about) abstract set-theoretic products. —David Eppstein (talk) 16:18, 22 October 2014 (UTC)

## Set of ALL ordered pairs

In the first paragraph we read: "for sets A and B, the Cartesian product A × B is the set of ALL ordered pairs (a, b) where a ∈ A and b ∈ B". But in my understanding this sentence is confusing and in contradiction to non-commutativity and non-associativity. I propose removal of the "all" quantifier because it suggests that for A = {1,2}; B = {3,4} the cartesian product is A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4), (3,1), (3,2), (4,1), (4,2)} - 86.5.220.129 (signature supplied from history)

The definition in the first paragraph is ok. In your example, e.g. (3,1) is not an ordered pair (a,b) where a∈A and b∈B, so it is not in A×B. - Jochen Burghardt (talk) 13:52, 1 August 2015 (UTC)