Necessary and Sufficient Conditions

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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.

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