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In logic, necessity and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement: 'If P then Q', Q is necessary for P' because P cannot be true unless Q is true. Similarly, 'P is sufficient for Q' because P being true always implies that Q is true, but P not being true does not always imply that Q is not true. In logic, necessity and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement: 'If P then Q', Q is necessary for P' because P cannot be true unless Q is true. Similarly, 'P is sufficient for Q' because P being true always implies that Q is true, but P not being true does not always imply that Q is not true. The assertion that a 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 only if N', 'S implies N', 'N is implied by S', S → N , S ⇒ N, or 'N whenever S'. In the above situation, we 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 is to be true (see third column of '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. In the above situation, we can also say S is a sufficient condition for N. Again, consider the third column of the truth table immediately below. 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 S ⇒ N {displaystyle SRightarrow N} and N ⇒ S {displaystyle NRightarrow S} (which can also be written as S ⇐ N {displaystyle SLeftarrow N} ) 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 S ⇔ N {displaystyle SLeftrightarrow N} . 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.

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