# Cartesian product (without order) and operation

I think that I know the formal definition of the Cartesian product of order pairs, but in my example the order doesn’t make sense. Let’s look at an example. I have 4 sets: $A_1=\{ milk, 3\}$, $A_2=\{water, 1\}$, $A_3=\{black, no, 1 \}$, $A_4=\{milk, no \}$

The Cartesian product $A = A_1 \times A_2 \times A_3 \times A_4 = \{$

$( milk , water , black , milk )$,

$( milk , water , black , no )$,

$( milk , water , no , milk )$,

$( milk , water , no , no )$,

$( milk , water , 1 , milk )$,

$( milk , water , 1 , no )$,

$( milk , 1 , black , milk )$,

$( milk , 1 , black , no )$,

$( milk , 1 , no , milk )$,

$( milk , 1 , no , no )$,

$( milk , 1 , 1 , milk )$,

$( milk , 1 , 1 , no )$,

$( 3 , water , black , milk )$,

$( 3 , water , black , no )$,

$( 3 , water , no , milk )$,

$( 3 , water , no , no )$,

$( 3 , water , 1 , milk )$,

$( 3 , water , 1 , no )$,

$( 3 , 1 , black , milk )$,

$( 3 , 1 , black , no )$,

$( 3 , 1 , no , milk )$,

$( 3 , 1 , no , no )$,

$( 3 , 1 , 1 , milk )$,

$( 3 , 1 , 1 , no ) \}$

Is the above right? This is my first question but next refers to operations. Let’s take a subset $$B_1=\{(milk,1,1,milk) \}, B_2=\{ (milk,1 ,1,no)\}, B_3=\{(milk,1,1,milk), (milk,1,1,no) \}$$ I think that every tuple out of 24 in set A is an element that doesn't change at all. I mean that $C_1=\{ (milk,1 ,1,no)\}$ and $C_2=\{ milk,1 ,1,no\}$ differs because $C_2=\{ milk,1,no\}$. If so:
$$B_1 \cap B_2 = \emptyset; B_1 \cup B_2 = B_3; B_1 \cap B_3 = B_1; B_1 - B_2=B_1; B_3 - B_2=B_2; B_1 - B_3= \emptyset; B_1 \cup B_2 \cup B_3 = B_3$$

Does anyone could confirm (or not) operations truthfulness?

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Yes, what you have written appears to be the the Cartesian product $A_1 \times A_2 \times A_2 \times A_4$.

(By the way, when you say "the formal definition of the Cartesian product of order pairs" I assume that you mean "the formal definition of the Cartesian product as a set of ordered pairs" and moreover that you are extending the definition in the natural way to define a 4-fold Cartesian product as a set of ordered 4-tuples.)

Your statements about your subsets $B_1$, $B_2$, and $B_3$ of the Cartesian product are all correct except that $B_3 - B_2$ is $B_1$, not $B_2$.

I'm not sure what you mean by "the order doesn’t make sense," so if it still doesn't make sense, please explain why. (Note that the Cartesian product does not come with any prescribed order—you could re-arrange your ordered 4-tuples within the curly braces and what you wrote would still be valid.)

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Trevor: thank you. You are absolutely right about order. Order within every 4-tuples (in the round braces) is crucial, but re-arranging 4-tuples within the curly braces can be done without limits. The last: of course $B_1$ - my stupid mistake. – user65517 Mar 7 '13 at 20:50