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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