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A "total order" of a set requires certain properties of the ordering function. For any A, B and C:

Transitivity: A>B and B>C implies A>C

Trichotomy: A>B or B>A or A=B

These properties are true of the '>' operator meaning "greater than" when used to compare real numbers. This means that real numbers can be put in order by comparing them in pairs to see which is greater.

Side note: without "Trichotomy", we would have a "partial order", where the order of the set would not be unique. For example, if the set were people, and '>' meant "is an ancestor of", then Transitivity would still be true, but Trichotomy would not. And there would be many ways to order a group of people so that descendants always came before ancestors.

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Oma Aufderhar

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โˆ™ 2022-05-23 16:45:56
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Q: How does comparing numbers help order numbers?
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