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).
Read more about this topic: Necessity And Sufficiency
Famous quotes containing the words relationship between, relationship, necessity and/or sufficiency:
“We must introduce a new balance in the relationship between the individual and the government—a balance that favors greater individual freedom and self-reliance.”
—Gerald R. Ford (b. 1913)
“Guilty, guilty, guilty is the chant divorced parents repeat in their heads. This constant reminder remains just below our consciousness. Nevertheless, its presence clouds our judgment, inhibits our actions, and interferes in our relationship with our children. Guilt is a major roadblock to building a new life for yourself and to being an effective parent.”
—Stephanie Marston (20th century)
“I complacently accepted the social order in which I was brought up. I probably would have continued in my complacency if the happy necessity of self-support had not fallen to my lot; if self-support had not deepened and widened my contacts and my experience.”
—Mary Barnett Gilson (1877–?)
“Our age is very cheap and intelligible. Unroof any house, and you shall find it. The well-being consists in having a sufficiency of coffee and toast, with a daily newspaper; a well glazed parlor, with marbles, mirrors and centre-table; and the excitement of a few parties and a few rides in a year.”
—Ralph Waldo Emerson (1803–1882)