In logic, the words necessity and sufficiency refer to the implicational relationships between statements. Logic is the study of the principles of valid demonstration and Inference. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true. ↔
The assertion that P is necessary for Q is colloquially equivalent to "Q cannot be true unless P is true. " By contraposition, this is the same thing as "whenever Q is true, so is P". For contraposition in the field of traditional logic see Contraposition (traditional logic. The logical relation between them is expressed as "If Q then P" and denoted "Q P" (Q implies P), and may also be expressed as any of "P, if Q," "P whenever Q," and "P when Q. In Logic and Mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every " One often finds, in mathematical prose for instance, several necessary conditions which, taken together, constitute a sufficient condition, as shown in Example 3.
Example 1: Consider thunder, technically the acoustic quality demonstrated by the shock wave that inevitably results from any lightning bolt in the atmosphere. It may fairly be said that thunder is necessary for lightning, since lightning cannot occur without thunder, too, occurring. That is, if lightning does occur, then there is thunder.
Example 2: Being at least 30 years old is necessary for serving in the U. S. Senate. If you are under 30 years old then it is impossible for you to be a senator. That is, if you are a senator, it follows that you are at least 30 years old.
Example 3: In algebra, in order for some set S together with an operation to form a group, it is necessary that be associative. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, associativity is a property that a Binary operation can have It is also necessary that S include a special element e such that for every x in S it is the case that e x and x e both equal x. It is also necessary that for every x in S there exist a corresponding element x” such that both x x” and x” x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of
To say that P is sufficient for Q is to say that in and of itself, knowing P to be true is adequate grounds to conclude that Q is true. The logical relation is expressed as "If P then Q" or "P Q," and may also be expressed as "P implies Q. " Several sufficient conditions may, taken together, constitute a single necessary condition, as illustrated in example 3.
Example 1: An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt.
Example 2: A U. S. president's signing a bill that Congress passed is sufficient to make the bill law, regardless of the fact that even in the event of a presidential veto it still could have become law through a congressional override. A veto, Latin for "I forbid" is used to Denote that a certain party has the right to stop unilaterally a certain piece of Legislation. In the United States Congress can override a presidential Veto by having a two-thirds majority vote in both the House of Representatives and Senate
Example 3: That the center of a playing card should be marked with a single large spade (♠) is sufficient for the card to be an ace. Three other sufficient conditions are that the center of the card be marked with a diamond (♦), heart (♥), or club (♣), respectively. None of these conditions is necessary to the card's being an ace, but their disjunction is, since no card can be an ace without fulfilling at least (in fact, exactly) one of the conditions.
Mathematically speaking, necessity and sufficiency are dual to one another. In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. First, for any statements P and Q, the assertion that "P is sufficient for Q" is the same as "Q is necessary for P", for both statements mean that P implies Q. Another facet of this duality is that, as illustrated above, conjunctions of necessary conditions may achieve sufficiency, while disjunctions of sufficient conditions may achieve necessity. For a third facet, identify every mathematical predicate P with the set S(P) of objects for which P holds true; then asserting the necessity of P for Q is equivalent to claiming that S(P) is a superset of S(Q), while asserting the sufficiency of P for Q is equivalent to claiming that S(P) is a subset of S(Q). In Mathematics, a predicate is either a relation or the Boolean-valued function that amounts to the Characteristic function or the
To say that P is necessary and sufficient for Q is to say two things, that P is necessary for Q and that P is sufficient for Q. ↔ Of course, it may instead be understood to say a different two things, namely that each of P and Q is necessary for the other. And it may be understood in a third equivalent way: as saying that each is sufficient for the other. One may summarize any—and thus all—of these cases by the statement "P if and only if Q," which is denoted by P Q.
For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Cycle in Graph theory and Computer science has several meanings A closed walk with repeated vertices allowed Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and vice versa. A philosopher might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in intension, they have identical extension. Not to be confused with the homophone Intention; or the related concept of Intentionality. In any of several studies that treat the use of signs for example in Linguistics, Logic, Mathematics, Semantics, and Semiotics, the "