Changes

The notation is a little awkward, but the data of Table&nbsp;A3 should make the sense clear.  The important thing to observe is that <math>\operatorname{E}_{ij}</math> has the effect of transforming each proposition <math>f : U \to \mathbb{B}</math> into a proposition <math>f^\prime : U \to \mathbb{B}.</math>  As it happens, the action of each <math>\operatorname{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \operatorname{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions.  Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\operatorname{T}_{00}, \operatorname{T}_{01}, \operatorname{T}_{10}, \operatorname{T}_{11},</math> to bear in mind their transformative character, or nature, as the case may be.  Abstractly viewed, this group of order four has the following operation table:

The notation is a little awkward, but the data of Table&nbsp;A3 should make the sense clear.  The important thing to observe is that <math>\operatorname{E}_{ij}</math> has the effect of transforming each proposition <math>f : U \to \mathbb{B}</math> into a proposition <math>f^\prime : U \to \mathbb{B}.</math>  As it happens, the action of each <math>\operatorname{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \operatorname{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions.  Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\operatorname{T}_{00}, \operatorname{T}_{01}, \operatorname{T}_{10}, \operatorname{T}_{11},</math> to bear in mind their transformative character, or nature, as the case may be.  Abstractly viewed, this group of order four has the following operation table:

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

| align="center" |

 

{| align="center" cellpadding="0" 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%"

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

|- style="height:50px"

|         %         |          |          |          |

| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>

|   ·    %   T_00  |  T_01  |  T_10  |  T_11  |

| width="22%" style="border-bottom:1px solid black" |

|         %         |          |          |          |

| width="22%" style="border-bottom:1px solid black" |

|         %         |          |          |          |

|   T_00  %   T_00  |   T_01  |   T_10  |   T_11  |

| width="22%" style="border-bottom:1px solid black" |

|         %          |         |         |         |

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

| width="22%" style="border-bottom:1px solid black" |

|         %          |         |         |         |

|   T_01  %  T_01  |  T_00  |  T_11  |  T_10  |

|- style="height:50px"

|         %          |          |          |          |

| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>

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

| <math>\operatorname{T}_{00}</math>

|         %          |         |         |         |

| <math>\operatorname{T}_{01}</math>

|   T_10  %  T_10  |  T_11  |  T_00  |  T_01  |

| <math>\operatorname{T}_{10}</math>

|         %          |          |          |          |

| <math>\operatorname{T}_{11}</math>

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

|- style="height:50px"

|         %          |         |         |          |

| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>

|   T_11  %  T_11  |  T_10  |  T_01  |  T_00  |

| <math>\operatorname{T}_{01}</math>

|         %          |          |          |          |

| <math>\operatorname{T}_{00}</math>

| <math>\operatorname{T}_{11}</math>

</pre>

| <math>\operatorname{T}_{10}</math>

|- style="height:50px"

| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>

| <math>\operatorname{T}_{10}</math>

| <math>\operatorname{T}_{11}</math>

| <math>\operatorname{T}_{00}</math>

| <math>\operatorname{T}_{01}</math>

|- style="height:50px"

| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>

| <math>\operatorname{T}_{11}</math>

| <math>\operatorname{T}_{10}</math>

| <math>\operatorname{T}_{01}</math>

| <math>\operatorname{T}_{00}</math>

|}

|}

It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is Klein's four-group <math>V_4\!</math> (from German ''Vier''), which it is.

 

It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is Klein's four-group <math>V_4\!</math> (from German ''Vier''), which this is.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!