Necessary and sufficient condition

Necessary conditions

The assertion that P is necessary for Q is colloquially equivalent to "Q cannot be true unlessP is true," or "if P is false then Q is false." By contraposition, this is the same thing as "whenever Q is true, so isP". The logical relation between them is expressed as "If Q thenP" 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." One often finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5.

Example 1: In order for it to be true that "John is a bachelor," it is necessary that it be also true that he is

1. unmarried
2. male
3. adult

since to state "John is a bachelor" implies John has each of those three additionalpredicates.

Example 2: 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.

Example 3: Consider thunder, in the technical sense, 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 4: Being at least 30 years old is necessary of 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 5: In algebra, in order for some set S together with an operation to form agroup, it is necessary that beassociative. It is also necessary that S include a special element e such that for every x in S it is the case that ex and x e both equal x. It is also necessary that for every x inS there exist a corresponding element x” such that both xx” and x” x equal the special element e. None of these three necessary conditions by itself is sufficient, but theconjunction of the three is.

Sufficient conditions

To say that P is sufficient for Q is to say that, in and of itself, knowingP to be true is adequate grounds to conclude that Q is true. (It is to say, at the same time, that knowingP not to be true does not, in and of itself, provide adequate grounds to conclude thatQ is not true, either.) The logical relation is expressed as "If P thenQ" 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 5.

Example 1: Stating that "John is a bachelor" implies that John is male. So knowing that it is true that John is a bachelor is sufficient to know that he is a male.

Example 2: A number's being divisible by 4 is sufficient (but not necessary) for its being even, but being divisible by 2 is both sufficient and necessary.

Example 3: 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 4: A U.S. president's signing a bill that Congress passed is sufficient to make the bill law. Note that the case whereby the president did not sign the bill, e.g. through exercising a presidentialveto, does not mean that the bill has not become law (it could still have become law through a congressionaloverride).

Example 5: 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.

怎么去理解他们。

必要条件是一系列的小“点”，或者说，必要条件是得到一个结果的条件。而充分条件是一个巨大的“点”。或者说，充分条件是一群小点的集合。或者说，充分条件是一个结果。

比如 人类呼吸 和 人类生存

人类呼吸是人类生存的一点，人类必须呼吸才能生存。因此，呼吸对于人类是必要的。人类呼吸对于人类生存是必要条件。人类生存对于人类呼吸是充分条件。

那么，当且仅当，充分条件和必要条件是1对1的时候，才是充分必要条件。

人类生存不能仅仅依靠呼吸，所以人类生存和呼吸并不是充分必要条件。

那么

比如：

他被我打了

我打了他

这样就可以说是充分必要条件。因为互相都是对方的条件，也都是对方的结果。他们是相等的。

个人理解恩。