Two sets (or ideas, or principles, or entities, or categories, etc.) are considered properly dichotomous iff: 1. They are mutually exclusive - that is, there is absolutely no overlap between them, and 2. Together, they are exhaustive - that is, there is nothing that is outside both of them. For example, within the Real numbers, "rational" and "irrational" form a proper dichotomy. That means that every Real number is either rational or irrational, and no number is both rational and irrational. For a counterexample, within the natural numbers, "prime" and "composite" are not a proper dichotomy. This is because the number 1 is neither prime nor composite. Notice, also, that dichotomies may be proper in one context, but not in another. For instance, in the set of integers, "even" and "odd" form a proper dichotomy. But in the realm of Real numbers, they don't. (Pi, for instance, is neither odd nor even.) Many of the old ...