# Necessary and Sufficient Conditions

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Necessary and Sufficient Conditions

Conditions in the absence of which assertion A obviously cannot be true are called necessary conditions for the correctness of assertion A, and conditions in the presence of which assertion A is obviously true are called sufficient conditions for the correctness of assertion A. For example, a necessary condition for the divisibility of an integer by 2 is that the number, if written in decimal system notation, does not end in 7. This condition is necessary but not sufficient, since, for example, the number 23 does not end in 7 but is nevertheless not divisible by 2. A sufficient condition for the divisibility of a number by 2 is that it end in zero. This condition is sufficient but not necessary, since the number 38 does not end in a zero but is nevertheless divisible by 2. The usual indication of divisibility by 2—for a number to be divisible by 2 it is necessary and sufficient that its last digit be divisible by 2—is an example of a condition that is simultaneously necessary and sufficient. The expression “necessary and sufficient” is often replaced by the expression “if and only if.”

Necessary and sufficient conditions are of great cognitive value. It is sometimes extraordinarily difficult to find the necessary and sufficient conditions that are suitable for use in complex mathematical problems. In such cases, attempts are made to broaden sufficient conditions as much as possible, that is, to make them include the greatest possible number of cases in which the fact which interests us holds, and to narrow the necessary conditions as much as possible, that is, to make them include the fewest possible superfluous cases, cases in which the given fact does not hold. Thus, sufficient conditions gradually approach necessary conditions. A typical classical example of this type of investigation are studies on convergence conditions for series.

References in periodicals archive ?
Now we introduce the new classes of generalized second-order hybrid invex functions which seem to be application-oriented to developing a new optimality-duality theory for nonlinear programming based on second-order necessary and sufficient optimality conditions.
The main purpose of this current note is to introduce a Hypergeometric distribution series and obtain necessary and sufficient conditions for this series belonging to the classes ( , ) T and ( , ) C .
Motivated by results on connection between various tubclasses of analytic functions by using the hypergeometric function by many authors particularly the authors (see[5- 10]), S.Porrwal [4] obtained the necessary and sufficient conditions for a function ( , ) F m z defined by using the poisson distribution belong to the class ( , ) T and ( , ) C .
The team interpreted these results to mean that "overtranscription of Cyp6g1 alone is both necessary and sufficient for P450-mediated DDT resistance."
Katz, 1972; see also Komatsu, 1992), which maintained that concepts are mentally represented as definitions, that is in terms of both necessary and sufficient features.
More specifically, definitions imply that concepts have both necessary and sufficient features.
In complex instances when these necessary and sufficient conditions are combined into single event trees, the probability of the outcome can be expressed mathematically as a function of the several originating conditions.
Thus, the event structure may be reduced to the single necessary and sufficient condition of full Soviet support for a regime of this type to ensure its perpetuation.
Relying largely upon a comparison of the bacteria whose full gene sets, or genomes, have been laid bare, two scientists now conclude that a mere 256 or so genes may be necessary and sufficient for the modern cell.
(Owens dispenses with the common-sense idea that coincidences have to be out-of-the-ordinary.) The notion of 'Causal factors' is then independently analysed in terms of necessary and sufficient conditions (plus laws of nature).
Of course, Owens has his own more technical account of coincidence, in terms of the independence of the necessary and sufficient conditions for the occurrence of the constituent events.
Owens adopts a philosophical regimentation of this concept, according to which a coincidence is a conjunction of two or more events, each of which has independent see of necessary and sufficient conditions (p.
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