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Simple question, please help

  1. Jun 26, 2005 #1
    ehm, sorry, i am a beginner in set theory. learning on my own.
    for cartesian product ordered pair, for example
    [itex] A = \{a_1, a_{2}, a_{3}\} \\
    B = \{b_{1}, b_{2}, b_{3}\} [/itex]

    is the product [itex] A \times B = \{a_{1}b{1}, a_{1}b{2}, a_{1}b{3}, a_{2}b{1}, \\
    a_{2}b{2}, a_{2}b{3}, a_{3}b{1}, a_{3}b{2}, a_{3}b{3},\} [/itex] ??

    What does [itex] A \times B = \{(a,b)\mid a \in A and b \in B\} [/itex] mean in detail?

    thanks.
     
  2. jcsd
  3. Jun 26, 2005 #2
    sorry, a bit mistyped the question above.

    for cartesian product ordered pair, for example
    [itex] A = \{a_1, a_2, a_3\} [/itex] and [itex]B = \{b_1, b_2, b_3\} [/itex]

    is the product [itex] A \times B = \{a_{1}b{1}, a_{1}b{2}, a_{1}b{3}, a_{2}b{1}, \\
    a_{2}b{2}, a_{2}b{3}, a_{3}b{1}, a_{3}b{2}, a_{3}b{3},\} [/itex] ??

    What does [itex] A \times B = \{(a,b)\mid a \in A \ and \ b \in B\} [/itex] mean in detail
    in terms of individual set member for this simple example?

    thanks.
     
  4. Jun 26, 2005 #3

    matt grime

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    the elements in the product are the pairs (a_i,b_j) for 1<= i,j <=3.

    what does a_1b_1 even mean?

    the product is all odered pairs (a,b) where a is in A and b is in B. nothing more nothing less.
     
  5. Jun 26, 2005 #4
    [itex] a_i [/itex] and [itex] b_i [/itex] are numbers or element such as 1, 6, 8, etc.

    i think i may understand your simple answer.

    thanks
     
  6. Jun 26, 2005 #5

    matt grime

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    the product of sets does not involve multiplying the elements; elements of sets do not necessarily even possess a multiplicationwhat if A were the set of results of drawing a card and B were the set of results of tossing a coin? if a were the three of diamonds and b heads, then what does ab mean?
     
  7. Jun 26, 2005 #6

    HallsofIvy

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    Notice the parentheses in [itex] A \times B = \{(a,b)\mid a \in A \ and \ b \in B\} [/itex]?

    What you want is
    [itex] A \times B = \{(a_{1},b{1}), (a_{1},b_{2}), (a_{1},b_3}), (a_{2},b_{1}), (a_{2},b_{2}),\\ (a_{2},b_{3}), (a_{3},b_{1}), (a_{3},b_{2}),(a_{3},b_{3}),\}[/itex]
     
    Last edited: Jun 26, 2005
  8. Jun 26, 2005 #7
    yup. now i understand it. thanks guys.
     
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