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To that purpose, I will set forth a way of thinking about relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite.
To that purpose, I will set forth a way of thinking about relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite.
For example, suppose that we are given the relations ''L'' ⊆ ''X'' × ''Y'', ''M'' ⊆ ''Y'' × ''Z''. Table 3 and Figure 4 present a couple of ways of picturing the constraints that are involved in constructing the relational composition ''L'' o ''M'' ⊆ ''X'' × ''Z''.
For example, suppose that we are given the relations ''L'' ⊆ ''X'' × ''Y'', ''M'' ⊆ ''Y'' × ''Z''. Table 3 and Figure 4 present a couple of ways of picturing the constraints that are involved in constructing the relational composition ''L'' o ''M'' ⊆ ''X'' × ''Z''.
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<pre>
| L o M # X | | Z |
| L o M # X | | Z |
o---------o---------o---------o---------o
o---------o---------o---------o---------o
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The way to read Table 3 is to imagine that you are
The way to read Table 3 is to imagine that you are playing a game that involves placing tokens on the squares of a board that is marked in just this way. The rules are that you have to place a single token on each marked square in the middle of the board in such a way that all of the indicated constraints are satisfied. That is to say, you have to place a token whose denomination is a value in the set ''X'' on each of the squares marked "''X''", and similarly for the squares marked "''Y''" and "''Z''", meanwhile leaving all of the blank squares empty. Furthermore, the tokens placed in each row and column have to obey the relational constraints that are indicated at the heads of the corresponding row and column. Thus, the two tokens from ''X'' have to denominate the very same value from ''X'', and likewise for ''Y'' and ''Z'', while the pairs of tokens on the rows marked "''L''" and "''M''" are required to denote elements that are in the relations ''L'' and ''M'', respectively. The upshot is that when just this much is done, that is, when the ''L'', ''M'', and !1! relations are satisfied, then the row marked "''L'' o ''M''" will automatically bear the tokens of a pair of elements in the composite relation ''L'' o ''M''.
playing a game that involves placing tokens on the
squares of a board that is marked in just this way.
The rules are that you have to place a single token
on each marked square in the middle of the board in
such a way that all of the indicated constraints are
satisfied. That is to say, you have to place a token
whose denomination is a value in the set X on each of
the squares marked "X", and similarly for the squares
marked "Y" and "Z", meanwhile leaving all of the blank
squares empty. Furthermore, the tokens placed in each
row and column have to obey the relational constraints
that are indicated at the heads of the corresponding
row and column. Thus, the two tokens from X have to
denominate the very same value from X, and likewise
for Y and Z, while the pairs of tokens on the rows
marked "L" and "M" are required to denote elements
that are in the relations L and M, respectively.
The upshot is that when just this much is done,
that is, when the L, M, and !1! relations are
satisfied, then the row marked "L o M" will
automatically bear the tokens of a pair of
elements in the composite relation L o M.
Figure 4 shows a different way of viewing the same situation.
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o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 4. Relational Composition
Figure 4. Relational Composition
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