ONT Re: Theory Of Relations
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TOR. Note 2
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In view of the fact that X = {i, j, k} is a finite universe,
indeed, such a tiny universe, we can easily figure out how
many relations over X there are for any finite arity n
that you might care to name.
n = 1. A 1-adic relation is a subset of X^1 = X.
There are exactly 2^3 = 8 subsets of X.
So there are 8 1-adic relations over X.
n = 2. A 2-adic relation is a subset of X^2 = X x X.
There are 3 x 3 = 3^2 = 9 ordered 2-tuples in X^2.
So there are just 2^9 = 512 2-adic relations over X
n = 3. A 3-adic relation is a subset of X^3 = X x X x X.
There are 3 x 3 x 3 = 3^3 = 27 ordered 3-tuples in X^3.
So there are just 2^27 = 134217728 3-adic relations over X.
Like the man said:
| Of triadic Being the multitude of forms is so terrific that
| I have usually shrunk from the task of enumerating them ...
Jon Awbrey
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