Changes

For example, suppose that we are given the relations <math>L \subseteq X \times Y</math> and <math>M \subseteq Y \times Z.</math>  Table&nbsp;3 and Figure&nbsp;4 present two ways of picturing the constraints that are involved in constructing the relational composition <math>L \circ M \subseteq X \times Z.</math>

For example, suppose that we are given the relations <math>L \subseteq X \times Y</math> and <math>M \subseteq Y \times Z.</math>  Table&nbsp;3 and Figure&nbsp;4 present two ways of picturing the constraints that are involved in constructing the relational composition <math>L \circ M \subseteq X \times Z.</math>

{| align="center" cellspacing="6" width="90%"

| align="center" |

 

{| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"

Table 3.  Relational Composition

|+ '''Table 3.  Relational Composition'''

o---------o---------o---------o---------o

|-

|         #  !1!   |   !1!   |   !1!   |

| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | &nbsp;

o=========o=========o=========o=========o

| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>

|   L    #    X   |   Y   |         |

| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>

o---------o---------o---------o---------o

| style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math>

|   M    #        |   Y   |   Z   |

|-

o---------o---------o---------o---------o

| style="border-right:1px solid black" | <math>L\!</math>

| L o M #    X   |         |   Z   |

| <math>X\!</math>

| <math>Y\!</math>

</pre>

| &nbsp;

|-

| style="border-right:1px solid black" | <math>M\!</math>

| &nbsp;

| <math>Y\!</math>

| <math>Z\!</math>

|-

| style="border-right:1px solid black" | <math>L \circ M</math>

| <math>X\!</math>

| &nbsp;

| <math>Z\!</math>

|}

|}

The way to read Table&nbsp;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 <math>X\!</math> on each of the squares marked <math>{}^{\backprime\backprime} X {}^{\prime\prime},</math> and similarly for the squares marked <math>{}^{\backprime\backprime} Y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} Z {}^{\prime\prime},</math> 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 <math>X\!</math> have to denominate the very same value from <math>X,\!</math> and likewise for <math>Y\!</math> and <math>Z,\!</math> while the pairs of tokens on the rows marked <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} M {}^{\prime\prime}</math> are required to denote elements that are in the relations <math>L\!</math> and <math>M,\!</math> respectively.  The upshot is that when just this much is done, that is, when the <math>L,\!</math> <math>M,\!</math> and <math>\mathit{1}\!</math> relations are satisfied, then the row marked <math>{}^{\backprime\backprime} L \circ M {}^{\prime\prime}</math> will automatically bear the tokens of a pair of elements in the composite relation <math>L \circ M.</math>

The way to read Table&nbsp;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 <math>X\!</math> on each of the squares marked <math>{}^{\backprime\backprime} X {}^{\prime\prime},</math> and similarly for the squares marked <math>{}^{\backprime\backprime} Y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} Z {}^{\prime\prime},</math> 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 <math>X\!</math> have to denominate the very same value from <math>X,\!</math> and likewise for <math>Y\!</math> and <math>Z,\!</math> while the pairs of tokens on the rows marked <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} M {}^{\prime\prime}</math> are required to denote elements that are in the relations <math>L\!</math> and <math>M,\!</math> respectively.  The upshot is that when just this much is done, that is, when the <math>L,\!</math> <math>M,\!</math> and <math>\mathit{1}\!</math> relations are satisfied, then the row marked <math>{}^{\backprime\backprime} L \circ M {}^{\prime\prime}</math> will automatically bear the tokens of a pair of elements in the composite relation <math>L \circ M.</math>