## Relations & Their Relatives • Discussion 8

In discussing the “combinatorial explosion” of dyadic relations that takes off in passing from a universe of two elements to a universe of three elements, I made the following observation:

Looking back from the ascent we see that the two-point universe $\{ \mathrm{I}, \mathrm{J} \}$ manifests a type of formal degeneracy (loss of generality) compared with the three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$  This is due to the circumstance that the number of “diagonal” pairs, those of the form $\mathrm{A\!:\!A},$ equals the number of “off-diagonal” pairs, those of the form $\mathrm{A\!:\!B},$ so the two-point case exhibits symmetries that will be broken as soon as one adds another element to the universe.

There are two types of symmetry that we might be talking about in this setting and it behooves us to keep them distinctly in mind:

1. There is the symmetry exhibited by pairs of the form $\mathrm{A\!:\!A}$ versus the asymmetry exhibited by pairs of the form $\mathrm{A\!:\!B}.$
2. There is the number of pairs of the form $\mathrm{A\!:\!A}$ versus the number of pairs of the form $\mathrm{A\!:\!B}$ and whether those numbers are equal or not.

The type of symmetry (“sameness in measure”) motivating the above observation is the second type, where the number of pairs on the diagonal is equal to the number of pairs off the diagonal.  That is the symmetry that will be broken when we pass from the 2-point universe to the 3-point universe.

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