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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).
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.
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:
In the previous version, the section on sufficient conditions had reversed which is the condition and which is the thing obtained. If you do it this way, it's hard to tell at a glance the main difference between necessary and sufficient: necessary is obtained-->condition, while sufficient is condition-->obtained.
These restrictions are stronger than for the first fundamental theorem, with convexity of preferences and production functions a sufficient but not necessary condition. [ 5 ] [ 10 ] A direct consequence of the second theorem is that a benevolent social planner could use a system of lump sum transfers to ensure that the "best" Pareto efficient ...
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:
Only 4% of U.S. pet owners currently insure their pets, often due to persistent myths about how these policies work. Separate fact from fiction around pet policies — and how they can save you a ...
Thus, tr AA* = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form