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A condition can be either necessary or sufficient without being the other. For instance, being a mammal (N) is necessary but not sufficient to being human (S), and that a number is rational (S) is sufficient but not necessary to being a real number (N) (since there are real numbers that are not rational).
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:
Thus, the principle of detailed balance is a sufficient but not necessary condition for entropy increase in Boltzmann kinetics. These relations between the principle of detailed balance and the second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected to the Boltzmann H-theorem for polyatomic gases. [16]
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.
Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies =, …,,,
Leibniz test's monotonicity is not a necessary condition, thus the test itself is only sufficient, but not necessary. (The second part of the test is well known necessary condition of convergence for all series.) Examples of nonmonotonic series that converge are:
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 .
Cause-in-fact is determined by the "but for" test: But for the action, the result would not have happened. [1] (For example, but for running the red light, the collision would not have occurred.) The action is a necessary condition, but may not be a sufficient condition, for the resulting injury.