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A strict total order is a set $A$ with a binary relation $<$ on it, satisfying irreflexivity, transitivity and totality.

Could you give an example of a strict total order, please?

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2 Answers

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Well, here's a simple example. Consider the set $A=\{1,2,3\}.$ We'd like to put a strict order of some kind on this set--the natural choice (pun intended, for those who see it) is to make $1$ the least element and $3$ the greatest. Let's call our relation $\prec$, so we want $1\prec 2,$ $2\prec 3,$ and $1\prec 3$. That is, $$\prec\::=\bigl\{\langle 1,2\rangle,\langle 2,3\rangle,\langle 1,3\rangle\bigr\}.$$ This can readily be shown to be irreflexive, transitive, and total on $A$. We can also take the inverse relation $$\succ\::=\:\prec^{-1}\:=\bigl\{\langle 2,1\rangle,\langle 3,2\rangle,\langle 3,1\rangle\bigr\}$$ to get another total order on $A$, and there are $4$ other total orders on $A$, too. (Can you find them all?)

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The simplest example: the relation $<$ on the set $\{0, 1\}$, defined as usual by $0 < 1$.

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