## You are here

Homegeneralized Cartesian product

## Primary tabs

# generalized Cartesian product

Given any family of sets $\{A_{j}\}_{{j\in J}}$ indexed by an index set $J$, the *generalized Cartesian product*

$\prod_{{j\in J}}A_{j}$ |

is the set of all functions

$f\colon J\to\bigcup_{{j\in J}}A_{j}$ |

such that $f(j)\in A_{j}$ for all $j\in J$.

For each $i\in J$, the *projection map*

$\pi_{i}\colon\prod_{{j\in J}}A_{j}\to A_{i}$ |

is the function defined by

$\pi_{i}(f):=f(i).$ |

The generalized Cartesian product is the product in the category of sets.

The axiom of choice is the statement that the generalized Cartesian product of nonempty sets is nonempty. The generalized Cartesian product is usually called the Cartesian product.

Defines:

projection map

Related:

CartesianProduct, ProductTopology, AxiomOfChoice, OrderedTuplet, FunctorCategory2

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Cartesian product

Why the generalized Cartesian product is here denoted by using the \prod (= upper-case pi or russian "p") symbol? I think it does not tally with the cross symbol used in Cartesian product of two sets. I have seen that some textbooks and some teachers in university use the more logical \bigtimes symbol. Is the cause of the PM practice that the PM LaTeX does not yet contain the \bigtimes symbol?

## Re: Cartesian product

It is used here because that is the standard mathematical notation. Also, it's similarr to using Sigma (\sum) to denote addition instead a big cross. After all.. you're "multiplying" spaces, so the usual notation from calculus is kind of sensical.

The product symbol (\product (the large pi)) dates back to Descartes (http://members.aol.com/jeff570/operation.html)

And on your comment, I've NEVER seen any textbook nor teacher in university that ever uses a large X symbol to denote product (just as I've not seen used a large + symbol instead of sum.

Incidentally, I don't think \bigtimes and \bigplus are standard latex symbols (at least I don't recall). Indeed, there are \bigotimes and \bigoplus, but they carry some meaning usually different than cartesian product or sum (they being usually tensor product and direct sum)

I hope this clears the doubt on why \product is used to denote arbitrary products of things instead of a large X

It's how it's been done in mathematics for long long time

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: Cartesian product

Thanks for your answer. Does it mean that we now should denote the Cartesian product of A and B with a \cdot?

Jussi

## Re: Cartesian product

I think the best answer to this question is the one that Cam gave me when I complained about how nice it would be if mathematicians would use a more precise and logical notation for functions such as lambda-notation --- trying to convert mathematicians to change their notations is "About on the same level as replacing English with Esperanto for the majority of mathematical papers."

The "Pi" notation is very widespread, so it'll be a good long while before it changes (if it does). Of course, that shouldn't stop you from using your notation. If another notation is more precise and logical than standard, notation, by all means use it. Just don't hold your breath waiting for the rest of the mathematical world to catch up.

Ray

## Re: Cartesian product

hehe no.. but it explains why we write A^2 instead of A\times A

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: Cartesian product

I'd just like to point out that I have seen quite a few books which use the large Pi for product in a ring, and large X for the generalised Cartesian product.

One particular professor who I've had, and who finds a lot of use for both of these symbols in his lectures, is David Jackson (http://www.math.uwaterloo.ca/~dmjackson/)

Both show up quite often in combinatorics, as you have Cartesian products on the level of sets, which are quite often transformed into iterated products in the ring of formal power series, and we actually write these differently.

It actually does help to distinguish formulas on the level of sets and in the ring quite nicely by presence of disjoint union and Cartesian product (X) signs, or summation and iterated product (Pi) signs.

- Cale Gibbard

## Re: Cartesian product

I guess it kinda makes sense to use different symbols when dealing with different kinds of products at once, but I still claim that the "\bigtimes" is much rare compared to \product which is used in most cases.

But as I noted also, in algebra (say in groups, modules, etc) usually directproduct (\oplus) is sometimes exchanged with \times

But returning to the original point (topological spaces), in tha particular concept I've always seen \product (PI) to denote arbitrary product of topological spaces

HOWEVER, at least in a particular instance I've seen a white box (like the ones at end of proofs) to denote product in the box topology

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: Cartesian product

oh my mistake, it was't about topological spaces but sets alone. But I still mention that it makes sense using \product (PI to denote arbitrary products (generalizing \times) just as \sum (SIGMA) denotes arbitrary sums (generalizing + )

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: Cartesian product

My teachers, some professors in the University of Turku, have already in 1970's used the big cross (\bigtimes) for denoting the Cartesian products in the group theory and the measure theory. (Also I when I lectured a course of abstract algebra in the University of Tampere in 1970 used this notation.) I still think, that it fits best together with the basic case AxB and makes clear distinction between the Cartesian product and the product in multiplication. But the force of habit determines the practice... =o(

## Re: Cartesian product

I have seen a book which use a big cross to denote Cartesian product. It is Foundations of Modern Probability by Professor Olav Kallenberg. It uses something like \Cross_k S_k frequently, but I fail to produce a big cross in my own tex file. In fact, there is an package called ifsym, which is said to produce the big cross and many other geometric symbols, but TeX keeps saying it doesn't know "\Cross" even though I have include ifsym package in the header of my tex file. It is very weird. I would be happy if anyone can help me get through this.

## Re: Cartesian product

Thanks! I have studied in University of Turku (Finland), and there several professors used the bigcross notation for Cartesian product, never the bigpi notation. In PM, I have not succeeded to make the bigpi =o(

Regards,

Jussi