Changes

{| 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>

|}

|}

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.

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.

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