# Talk:Trichotomy (mathematics)

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Field: Foundations, logic, and set theory

## Theology

I added the theological definition of trichotomy to the listing of definitions. meng.benjamin March 27, 2006 16:40 EST

## AC

Removed assertion that Trichotomy is equivalent to AC. As evidenced by the proof at http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem, this isn't true. 129.97.97.134 (talk) 00:30, 11 June 2008 (UTC)

No, Cantor-Bernstein doesn't imply trichotomy. Cantor-Bernstein says that cardinals are partially ordered. Trichotomy says that they are totally ordered. And this is indeed equivalent to AC, because it implies that every set is smaller than some ordinal (and can therefore be well-ordered). So I'm reverting your edit. --Zundark (talk) 16:37, 11 June 2008 (UTC)

## Boolean Expression

If there is a Boolean expression for trichotomy using AND, OR and NOT it would be much appreciated.

The "exclusiveness" of the Trichotomy can be represented using Exclusive OR (XOR). For example: xRy XOR yRx XOR x=y. But if you want to limit yourself only to AND, OR and NOT for some reason. XOR can be represented by some combinations of them as well. Paulmiko (talk) 23:12, 29 June 2011 (UTC)

## proof?

The sentence "The law of trichotomy was long assumed true without proof; it was proven true at the end of the 19th century.[1]" doesn't make much sence to me, exactely what has been proved?--Sandrobt (talk) 20:47, 11 October 2010 (UTC)

I agree, it doesn't make much sense. I suppose it was meant to say "the trichotomy of the order relation on real numbers was unproven...", still, I am not sure how this is verifiable. So I removed this sentence. ComputScientist (talk) 14:48, 30 January 2011 (UTC)

## Conflict: Trichotomy - Reflexivity

Hello everyone, the Trichotomy focuses on order relations, but: Don't they satisfy the reflexivity per definition? So consider the pairs (x,x), every case is true (xRx, xRx and x=x) and so the "excluding or" gives false.

Best regards -- 138.246.2.199 (talk) 03:55, 7 September 2011 (UTC)

Order relations are usually defined to be reflexive, but they can also be defined to be irreflexive (see Partially ordered set#Strict and non-strict partial orders for details). For trichotomy, it's the irreflexive form that should be used (which is why the article uses < and > rather than ≤ and ≥). --Zundark (talk) 07:56, 7 September 2011 (UTC)