Results 1 to 3 of 3

Math Help - Cardinality Of A Cartesian Product

  1. #1
    Member
    Joined
    May 2011
    Posts
    178
    Thanks
    6

    Cardinality Of A Cartesian Product

    Hello,

    I am having difficulty understanding why the cardinality of a Cartesian Product is simply the product of the cardinality of the individual sets involved in the Cartesian Product. As an attempt in trying to understand this, I supposed the set A x A, where A has n elements: I understand that 1 goes to 1, 1 goes to 2,..., and 1 goes to n, which can be stated as (1,1),(1,2),...,(1,n); furthermore, 2 goes to 1, 2 goes to 2,..., and 2 goes to n, which can be stated as (2,1), (2,2),...,(2,n). However, this train of thinking did not do an adequate job in elucidating this concept. Could someone please help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,792
    Thanks
    1687
    Awards
    1

    Re: Cardinality Of A Cartesian Product

    Quote Originally Posted by Bashyboy View Post
    I am having difficulty understanding why the cardinality of a Cartesian Product is simply the product of the cardinality of the individual sets involved in the Cartesian Product.
    First, I will assume that you are considering only finite sets.

    Look at the definition: A\times B=\{(x,y):\;x\in A~\&~y\in B\}.
    How many pairs are possible?
    If the cardinality of A is |A|=n then there are only ways n to pick the x value of the pairs.
    Likewise, there are only |B|=m ways to pick the y values in the pairs.

    Thus |A\times B|=|A|\cdot|B|=nm.

    Example: \{a,s,d,f,g\}\times\{1,2,3\} contains 15 pairs.

    Now for infinite sets, things are more complicated.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: Cardinality Of A Cartesian Product

    If you associate a number with each element of A you can lay out the elements of AXA as:

    11, 12, 13, 14, ..
    21, 22, 23, 24,


    And then start counting on a diagonal starting from upper left: 11,21,12,31,22,13. (1,2,,3,)

    Just something I remembered from an analysis text. Not sure Im convinced.

    EDIT: The reason I am not convinced is that if you started counting by row, you would never get past the first row.
    Last edited by Hartlw; April 12th 2013 at 08:50 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cartesian product of A*A*A*A
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 14th 2011, 06:29 AM
  2. Cartesian product
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 19th 2011, 11:38 AM
  3. Cartesian product.
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: November 4th 2010, 01:59 PM
  4. Cartesian Product
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 9th 2010, 11:31 AM
  5. Cartesian Product
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: November 15th 2009, 08:23 AM

Search Tags


/mathhelpforum @mathhelpforum