Multiplication, Cartesian Products, Product Sets, and Tuples

Tuples and cartesian products:

Imagine you have a set A and a set B. A={1,2,} and B={c,d}. A x B= {(1,c), (1,d),(2, c),(2,d)}. Now label each member of A a_1, a_2 and each member of B b_1, b_2. Now if you had A x A, that would be {(a_{11}, a_{21}), (a_{11}, a_{22}), (a_{12}, a_{21}), (a_{12}, a_{22})}. That’s basically all the permutations of the set A combined with the set A, ordered and with repetition, that’s A^n permutations where n is the number of A sets you’re multiplying.

“A particularly interesting extension of the idea of multiplication is in the Cartesian product of two sets. If A = {1, 2, 3}, and B = {x, y}, then A × B is the set {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)}, formed by pairing each element of A with each element of B. Because sets are sometimes used as the basis for arithmetic, Cartesian products form an important link between sets and ordinary multiplication.” (Gale Science Encyclopedia Vol 4)

If you label the members of A as a, and of B as b, then A x B will equal all ordered pairs (a,b). Note that for such tuples, tuples are surrounded by parentheses () while sets are surrounded by curly brackets {}.

“The Cartesian product of two sets A and B consists of all ordered pairs (a,b) where a is a member of A and b is a member of B.” (Wikipedia: Set Theory)

Also see:

  • …a tuple is a sequence (also known as an “ordered list”) of values…
  • The main properties that distinguish a tuple from, for example, a set are that
    • it can contain an object more than once;
    • the objects appear in a certain order;
    • it has finite size.
  • In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.
  • A sequence of a fixed-length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements.
  • Wikipedia: Sets
  • Wikipedia: Set Theory
  • Wikipedia: Cardinality
  • Wikipedia: Multiplication
  • Wolfram Math World: List
  • An data structure consisting of an ordered set of elements, each of which may be a number, another list, etc. A list is usually denoted (a_1, a_2, …, a_n) or {a_1,a_2,...,a_n}, and may also be interpreted as a vector (specifically, an n-vector) or an n-tuple. Multiplicity matters in a list, so (1, 1, 2) and (1, 2) are not equivalent.
  • Wolfram Math World: n-tuple
  • An n-tuple, sometimes simple called a “tuple” when the number n is known implicitly, is another word for a list, i.e., an ordered set of n elements. It can be interpreted as a vector, or more specifically, an n-vector.”
  • Open Learn: Learning Space: 2.6 Associations: tuples and Cartesian products
  • Consider an item of shopping that is weighed at the supermarket checkout, such as 335 grams of walnuts. This item of shopping has two features: the type of item purchased (walnuts), and the weight of that item (335 grams). To record a weighed item of shopping we need to note both these features. This can be done using an ordered pair: (“WALNUTS”, 335).

    The first item in this ordered pair gives the type of item purchased. Let WeighedItems be the set of items stocked by the supermarket that need to be weighed. The first item comes from this set, while the second item in the ordered pair comes from the set Int.

    We call the set containing all such pairs the Cartesian product of the two sets. (‘Cartesian’ after the famous French philosopher and mathematician René Descartes.) This set of pairs is written as WeighedItems × Int. So the set WeighedItems × Int consists of all pairs (w, n), where w comes from the set WeighedItems, and n comes from the set Int. We refer to (w, n) as an ordered pair. The word pair here indicates that there are two items grouped together. The word ordered indicates that the order in which the two items are given matters. The pair (“WALNUTS”, 335) is not the same as the pair (335, “WALNUTS”). Similarly, the set WeighedItems × Int is different from the set Int × WeighedItems, which consists of pairs giving: first an integer, then a type of weighed item.

    Use of the symbol × here has nothing to do with multiplication of numbers!

    We can form the Cartesian product of any two sets. The Cartesian product of sets X and Y is written as X × Y , and consists of all ordered pairs (x, y) where x is from the set X and y is from the set Y. X × Y can be read as “ X cross Y”…

  • …The terminology ‘ordered triple’ indicates that we are now writing three items, each from a specified set, in a given order. We may want to associate four items, or five, or more. The terminology ‘tuple’ extends conveniently to this general case, where we talk about 4-tuples, or 5-tuples, and, in general, n-tuples (where n might be 2, or 3, or any larger integer). The set of all 3-tuples of the same form as (13151, “ORGANIC SOUP”, 179) is again a Cartesian product, in this case a product of three sets.
  • Wikipedia: Vectors
  • A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment connecting the initial point A with the terminal point B and denoted
  • The magnitude is the length of the segment and the direction characterizes the displacement of B relative to A: how much one should move the point A to “carry” it to the point B.[1]

    One thing I’m not sure of yet is how to number n-tuples. Here’s a question I found on Yahoo Answers which I too would like to see answered:

    How do you number n-tuples/ordered lists/product sets?

    If you have say sets X_1, X_2, X_3 where the underscore denotes subscript, and the product set X_1 x X_2 x X_3, how do you number each of the ordered lists/3-tuples ( X_1, X_2, X_3 )? Would you number them like ( X_1, X_2, X_3 )_n,m,o, where n is the cardinality of the set X_1, where m is the cardinality of the set X_2, and where o is the cardinality of the set X_3? So if n=m=o for example, you would number them ( X_1, X_2, X_3 )_1, 1, 1, ( X_1, X_2, X_3 )_1, 1, 2, ( X_1, X_2, X_3 )_1, 1, 3, ( X_1, X_2, X_3 )_1, 2, 1, ( X_1, X_2, X_3 )_1, 2, 2, etc, until you get to ( X_1, X_2, X_3 )_3,3,3, where once again the underscore stands for subscript?

    Check out the answer so far: it seems that this type of notation is atypical…

That kind of confusing notation can be seen on p. 95, Exercise 10.2, question 4 in Norman Bigg’s Discrete Mathematics, much or most of which amazingly can be read for free online via Google Books! “If X1, X2, ….Xn are sets, the product set X1 x X2 x…Xn is defined to be the set of all ordered n-tuples (x1, x2, … xn)…” Another great math book available to read online via Google Books is Mathematics for Economists by Malcolm Pemberton and Nicholas Rau.

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