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| {| align="center" cellpadding="8" width="90%" | | {| align="center" cellpadding="8" width="90%" |
− | | <math>P \times Q \ = \ P \times X\ \cap\ X \times Q.</math> | + | | |
| + | <math>\begin{array}{lllll} |
| + | P \times Q & = & P \times X & \cap & X \times Q. \\ |
| + | \end{array}</math> |
| |} | | |} |
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| The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation. | | The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation. |
| | | |
− | <pre>
| + | In a corresponding fashion at the level of the elements, the ordered pair <math>(p, q)\!</math> is described by a conjunction of propositions, namely, <math>p_1 \land q_2,</math> subject to the following interpretation: |
− | In a corresponding fashion at the level of the elements, | |
− | the ordered pair <p, q> is described by a conjunction | |
− | of propositions, namely, "p_<1> and q_<2>", subject | |
− | to the following interpretation: | |
| | | |
− | 1. "p_<1>" says that p is in the first place
| + | # <math>p_1\!</math> says that <math>p\!</math> is in the first place of the product element under construction. |
− | of the product element under construction.
| + | # <math>q_2\!</math> says that <math>q\!</math> is in the second place of the product element under construction. |
− | | |
− | 2. "q_<2>" says that q is in the second place
| |
− | of the product element under construction.
| |
| | | |
| + | <pre> |
| Notice that, in construing the cartesian product of the sets P and Q or the | | Notice that, in construing the cartesian product of the sets P and Q or the |
| concatenation of the languages L_1 and L_2 in this way, one shifts the level | | concatenation of the languages L_1 and L_2 in this way, one shifts the level |