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Thread: Cartesian Product

  1. #1
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    Cartesian Product

    What is the meaning of product in the phrase "Cartesian Product"?
    Why we use the word product?
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  2. #2
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    Re: Cartesian Product

    Quote Originally Posted by policer View Post
    What is the meaning of product in the phrase "Cartesian Product"?
    Why we use the word product?
    Because the result is very similar to how multiplication works when distributed.

    $$a\times b = \left( \underbrace{1+1+\cdots +1}_{a\text{ 1's}} \right)b = \underbrace{1\cdot b+\cdots 1\cdot b}_{1\cdot b\text{ appears }a\text{ times}}$$

    This is similar to

    $$\{a_1, a_2, \cdots\} \times \{b_1, b_2, \cdots \} = \left\{ \underbrace{(a_1,b_1),(a_1,b_2),\ldots}_{a_1\text{ is on the left for every }b_i}, \underbrace{(a_2,b_1),\ldots}_{a_2\text{ is on the left for every }b_i}, \ldots \right\}$$
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  3. #3
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    Re: Cartesian Product

    And thinking more about this, another (simpler) reason it is called a product is because given finite sets $A,B$, the cardinality $|A\times B| = |A|\cdot |B|$.
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