Relationship Between Necessity and Sufficiency
A condition can be either necessary or sufficient without being the other. For instance, being a mammal (P) is necessary but not sufficient to being human (Q), and that a number q is rational (P) is sufficient but not necessary to q's being a real number (Q) (since there are real numbers that are not rational).
A condition can be both necessary and sufficient. For example, at present, "today is the Fourth of July" is a necessary and sufficient condition for "today is Independence Day in the United States." Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M has a nonzero determinant.
Mathematically speaking, necessity and sufficiency are dual to one another. For any statements P and Q, the assertion that "P is necessary for Q" is equivalent to the assertion that "Q is sufficient for P." Another facet of this duality is that, as illustrated above, conjunctions of necessary conditions may achieve sufficiency, while disjunctions of sufficient conditions may achieve necessity. For a third facet, identify every mathematical predicate P with the set S(P) of objects for which P holds true; then asserting the necessity of P for Q is equivalent to claiming that S(P) is a superset of S(Q), while asserting the sufficiency of P for Q is equivalent to claiming that S(P) is a subset of S(Q).
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Famous quotes containing the words relationship, necessity and/or sufficiency:
“Every relationship that does not raise us up pulls us down, and vice versa; this is why men usually sink down somewhat when they take wives while women are usually somewhat raised up. Overly spiritual men require marriage every bit as much as they resist it as bitter medicine.”
—Friedrich Nietzsche (1844–1900)
“Whoever is not in the possession of leisure can hardly be said to possess independence. They talk of the dignity of work. Bosh. True work is the necessity of poor humanity’s earthly condition. The dignity is in leisure. Besides, 99 hundredths of all the work done in the world is either foolish and unnecessary, or harmful and wicked.”
—Herman Melville (1819–1891)
“The worthiest man to be known, and for a pattern to be presented to the world, he is the man of whom we have most certain knowledge. He hath been declared and enlightened by the most clear-seeing men that ever were; the testimonies we have of him are in faithfulness and sufficiency most admirable.”
—Michel de Montaigne (1533–1592)