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The example indicates that less than is not a total relation over the set of real numbers because "one can pick two equal numbers." If we are dealing with the SET of real numbers, then by definition, only one of each number can exist in that set. Therefore two equal numbers could never be picked, and
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For example, "is less than or equal to" is a total relation over the set of real numbers, because for two numbers either the first is less than or equal to the second, or the second is less than or equal to the first. On the other hand, "is less than" is not a total relation, since one can pick two
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equal numbers, and then neither the first is less than the second, nor is the second less than the first. (But note that "is less than" is a weak order which gives rise to a total order, namely "is less than or equal to". The relationship between strict orders and weak orders is discussed at
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The use of this namespace for a type of relation is not supported by a reference. In fact, references are available that use total relation in the sense of
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The relation "is a subset of" is also not total because, for example, neither of the sets {1,2} and {3,4} is a subset of the other.
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Connexity does not imply reflexivity. A strict partial order is a strict total order if and only if it is connex.
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numbers have to be picked. One property of a total relation is that for every element
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These references confirm standard usage departing from contents of this article. —
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less than could be a total relation. Can anyone clarify this?
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