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. That is, the two statements must be either simultaneously true or simultaneously false.
In ordinary English, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother.
In the conditional statement, "if S, then N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent. This conditional statement may be written in many equivalent ways, for instance, "N if S", "S implies N", "S only if N", "N is implied by S", S ⇒ N, or "N whenever S".
In the above situation, we also say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement, then the consequent N must be true if S may at all be true (see "truth table" immediately below). Phrased differently, the antecedent S cannot be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named.
We also say that S is a sufficient condition for N. Consider the truth table again. If the conditional statement is true, then if S is true, N must be true; whereas if the conditional statement is true and N is true, then S may be true or be false. In common terms, "S guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that someone has a Name.
A necessary and sufficient condition requires that both of the implications and (which can also be written as ) hold. From the first of these we see that S is a sufficient condition for N, and from the second that S is a necessary condition for N. This is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or .
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". By contraposition, this is the same thing as "whenever P is true, so is Q".
The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q). It may also be expressed as any of "P only if Q", "Q, if P", "Q whenever P", and "Q when P". One often finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5.
If P is sufficient for Q, then knowing P to be true is adequate grounds to conclude that Q is true; however, knowing P to be false does not meet a minimal need to conclude that Q is false.
The logical relation is, as before, expressed as "if P, then Q" or "P ⇒ Q". This can also be expressed as "P only if Q", "P implies Q" or several other variants. It may be the case that several sufficient conditions, when taken together, constitute a single necessary condition, as illustrated in example 5.
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).
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". Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M has a nonzero determinant.
Mathematically speaking, necessity and sufficiency are dual to one another. For any statements S and N, the assertion that "N is necessary for S" is equivalent to the assertion that "S is sufficient for N". Another facet of this duality is that, as illustrated above, conjunctions (using "and") of necessary conditions may achieve sufficiency, while disjunctions (using "or") of sufficient conditions may achieve necessity. For a third facet, identify every mathematical predicate N with the set T(N) of objects, events, or statements for which N holds true; then asserting the necessity of N for S is equivalent to claiming that T(N) is a superset of T(S), while asserting the sufficiency of S for N is equivalent to claiming that T(S) is a subset of T(N).
To say that P is necessary and sufficient for Q is to say two things:
One may summarize any, and thus all, of these cases by the statement "P if and only if Q", which is denoted by , whereas cases tell us that is identical to .
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. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and conversely. 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.
In mathematics, theorems are often stated in the form "P is true if and only if Q is true". Their proofs normally first prove sufficiency, e.g. . Secondly, the opposite is proven,
This proves that the circles for Q and P match on the Venn diagrams above.
Because, as explained in previous section, necessity of one for the other is equivalent to sufficiency of the other for the first one, e.g. is equivalent to , if P is necessary and sufficient for Q, then Q is necessary and sufficient for P. We can write and say that the statements "P is true if and only if Q, is true" and "Q is true if and only if P is true" are equivalent.
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