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# The cartesian product of sets

Definition 3.3.1   Let and be two sets. The cartesian product of and , denoted , is the set of all ordered pairs such that and .

For example, if and , then

Graphic presentations can always help to understand; we represent the elements of as points on an horizontal axis and the elements of on a vertical axis, as in figure 18. The elements of the cartesian products are then represented by the crossing points of the lattice of parallels to the axes through the points defined above. In Figure , we display a graphical representation for the above example.

Remark 3.3.2   In general, .

For example, if and are as above, we have:

Notation 1   For any set , we denote .

The set studied extensively in Linear Algebra is the cartesian product of with itself. In this case the coordinate axes in the plane show the plane as a graphic presentation of . The notion of the cartesian product of two sets can be generalized to three (or more) sets. First note that for three sets , and , we have : the elements of are ordered pairs whose first component is an element of and whose second element is an ordered pair belonging to ; the elements of are ordered pairs whose first component is an ordered pair belonging to and whose second element belongs to . Now we denote by the set of ordered triples where , and .

We generalize this definition to the product of sets , , :

For a set , the cartesian power is defined in the obvious way.

Proposition 3.3.3   If , and are three sets, then

Proof. The pair is an element of if, and only if, and , i.e. and ; therefore and , i.e. . Conversely, take , i.e. and ; it follows that and , thus and .

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root 2002-06-10