This article discusses only the formal meanings of necessary and sufficient. For the causal meanings see causation. Causality (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence

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.

• A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement P is a necessary condition of a statement Q if Q implies P. For example, the ability to breathe is necessary to a human's survival. Likewise, for the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.
• A sufficient condition is one that, if satisfied, assures the statement's truth. Formally, a statement P is a sufficient condition of a statement Q if P implies Q. Thus, jumping is sufficient to leave the ground, since an intrinsic element of the concept jumping is leaving the ground. A number's being divisible by 2 is sufficient for its being even.
• That a condition is one of necessary and sufficient does not imply the other. For instance, being a mammal is necessary but not sufficient to being human, and that a number q is rational is sufficient but not necessary to q‘s being a real number. In Mathematics, the real numbers may be described informally in several different ways A condition can be both necessary and sufficient. For example, at present, "today is the Fourth of July" is a necessary and sufficient condition for "today is Independence Day in the United States. In the United States, Independence Day, commonly known as the Fourth of July (or the Fourth) is a Federal holiday commemorating the adoption An Independence Day is an annual celebration commemorating the Anniversary of a Nation 's assumption of independent Statehood, usually after ceasing The United States of America —commonly referred to as the " Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M have a nonzero determinant. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

## Necessary conditions

Lightning is both necessary and sufficient for thunder, and vice versa. Lightning is an atmospheric discharge of Electricity, which typically occurs during Thunderstorms and sometimes during volcanic eruptions or Thunder is the sound made by Lightning. Depending on the nature of the lightning and distance of the listener it can range from a sharp This is because the two events are part of the same phenomenon.

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 $\Rightarrow$ 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 $\star$ to form a group, it is necessary that $\star$ 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 $\star$ x and x $\star$ e both equal x. It is also necessary that for every x in S there exist a corresponding element x” such that both x $\star$ x” and x$\star$ 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

## Sufficient conditions

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 $\Rightarrow$ 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.

## Relationship between necessity and sufficiency

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

## Simultaneous necessity and sufficiency

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 $\Leftrightarrow$ Q.