Editing Necessity and sufficiency (section)

==Necessity==

[[File:Solar eclipse 1999 4.jpg|thumb|right|200px|The sun being above the horizon is a necessary condition for direct sunlight; but it is not a sufficient condition, as something else may be casting a shadow, e.g., the moon in the case of an [[solar eclipse|eclipse]].]]

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".<ref name=":2" /><ref name=":0" /> 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'' [[Logical consequence|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 (i.e., individually necessary and jointly sufficient<ref name=":2" />), as shown in Example 5.

;Example 1: For it to be true that "John is a bachelor", it is necessary that it be also true that he is
:# unmarried,
:# male,
:# adult,
:since to state "John is a bachelor" implies John has each of those three additional [[Predicate (mathematical logic)|predicates]].

;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, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is thunder. The thunder ''does not cause'' the lightning (since lightning causes thunder), but because lightning always comes with thunder, we say that thunder is necessary for lightning. (That is, in its formal sense, necessity doesn't imply causality.)

;Example 4: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 must be at least 30 years old.

;Example 5:In [[algebra]], for some [[Set (mathematics)|set]] ''S'' together with an [[Binary operation|operation]] <math>\star</math> to form a [[group (mathematics)|group]], it is necessary that <math>\star</math> be [[associative]]. It is also necessary that ''S'' include a special element ''e'' such that for every ''x'' in ''S'', it is the case that ''e'' <math>\star</math> ''x'' and ''x'' <math>\star</math> ''e'' both equal ''x''. It is also necessary that for every ''x'' in ''S'' there exist a corresponding element ''x&Prime;'', such that both ''x'' <math>\star</math> ''x&Prime;'' and ''x&Prime;'' <math>\star</math> ''x'' equal the special element ''e''. None of these three necessary conditions by itself is sufficient, but the [[conjunction (logic)|conjunction]] of the three is.