Why is a dictionary so easy to use, however large it may be? Because everything is neatly arranged! Alphabetical order saves us a great deal of trouble. There are many other situations in which we need to sort things. The latest Olympic Games provide examples galore.
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A little order…
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What do all these ways of ranking things have in common? Mathematicians have identified three key properties that define an order relation R on a set E:
> – R must be reflexive: for every element x of E, we have x R x;
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> – R must be transitive: if x R y and y R z, then x R z;
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> – finally, R must be antisymmetric: if x and y are distinct, then either x R y or y R x, but not both! In other words, if x R y and y R x, then x = y.
