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Let us investigate the powers of $<$:

  • ${<^1} = \{(0,1);(0,2);(0,3);...;(1;2);...\}$
  • ${<^2} = \{(0,2);(0,3);(0,4);...;(1;3);...\}$
  • ${<^3} = \{(0,3);(0,4);(0,5);...;(1;4);...\}$
  • ...
  • ${<^N} = \{(0,N);(0,N+1);(0,N+2);...;(1;N+1);...\}$
  • ${<^0} = \{(0,0);(0,1);(0,2);...;(1;1);...\} = {\le}$
  • $<^{-1} = \{(0,-1);(0,0);(0,1);...;(1;0);...\}\ne {>}$

But why?

egreg
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Xorter
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  • Did you try applying the definitions, instead of just playing around with numbers and making arbitrary (and wrong) inferences? – egreg Jul 10 '16 at 10:30
  • Yes, I did. But without any useful result... – Xorter Jul 10 '16 at 13:53
  • I don't think so, because the definition of $<^{-1}$ is rather clear. You're arbitrarily inferring for negative numbers what you have for positive numbers. That's really wrong. – egreg Jul 10 '16 at 13:56
  • And what is about the fractions or complexes? E.g. what is <^0.5 ? – Xorter Jul 10 '16 at 14:20
  • Unless you define it, it means nothing at all. – egreg Jul 10 '16 at 14:29

1 Answers1

2

You got it wrong. So, you're looking at a relation $<\,\, \in \mathbb{N}_0 \times \mathbb{N}_0$. By definition,

$$<^n \,\,:= \{(a_0, a_n) \mid \exists a_0, a_1, \ldots, a_n \in \mathbb{N}_0 : a_0 < a_1 < \ldots < a_n \}$$

for positive $n$, but

$$<^0 \,\,:= \{(a, a) \mid a \in \mathbb{N_0}\} = \,\,\,=$$

and

$$<^{-1} \,\,:= \{(b, a) \mid (a, b) \in \,\,<\} = \{(b, a) \mid a < b\} = \{(b, a) \mid b > a\} =\,\,>.$$

Note that these are entirely different definitions than positive powers of relations. So you can't just extrapolate for negative numbers.

Anon
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  • I know the definitions, but according to the nth power of the relation, I got a different thing from the definition. Why? – Xorter Jul 10 '16 at 10:28
  • The rule you found is only true for positive $n$. – Anon Jul 10 '16 at 10:28
  • And Is there a generally rule for every n? – Xorter Jul 10 '16 at 10:33
  • There's a rule for $n > 0$, a rule for $n = 0$, and a rule for $n < 0$. Combine those to get a piecewise general rule for all $n$. – Anon Jul 10 '16 at 10:37
  • @Xorter Based on the rule for positive $n$ and the rule for $-1$ the rule for negative $n$ is: $R^n=(R^{-1})^{-n}$, or (same result) $R^n=(R^{-n})^{-1}$. – drhab Jul 10 '16 at 10:42