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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.
It is a necessary condition that an object has four sides if it is true that it is a square; conversely, the object being a square is a sufficient condition for it to be true that an object has four sides. [4] Four distinct combinations of necessity and sufficiency are possible for a relationship of two conditions. A first condition may be:
If P and Q are "equivalent" statements, i.e. , it is possible to infer P under the condition Q. For example, the statements "It is August 13, so it is my birthday" P → Q {\displaystyle P\to Q} and "It is my birthday, so it is August 13" Q → P {\displaystyle Q\to P} are equivalent and both true consequences of the statement "August 13 is my ...
In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. [3]
The absence these conditions guarantees the outcome cannot occur, and no other condition can overcome the lack of this condition. Further, necessary conditions are not always sufficient. For example, AIDS necessitates HIV, but HIV does not always cause AIDS. In such instances, the condition demonstrates its necessity but lacks sufficiency.
The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum []. A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.
The modern [1] formulation of the principle is usually ascribed to early Enlightenment philosopher Gottfried Leibniz.Leibniz formulated it, but was not an originator. [2] The idea was conceived of and utilized by various philosophers who preceded him, including Anaximander, [3] Parmenides, Archimedes, [4] Plato and Aristotle, [5] Cicero, [5] Avicenna, [6] Thomas Aquinas, and Spinoza. [7]