Editing Necessity and sufficiency (section)

==Definitions==

In the conditional statement, "if ''S'', then ''N''", the expression represented by ''S'' is called the [[Antecedent (logic)|antecedent]], and the expression represented by ''N'' is called the [[consequent]]. This conditional statement may be written in several equivalent ways, such as "''N'' if ''S''", "''S'' only if ''N''", "''S'' implies ''N''", "''N'' is implied by ''S''", {{math|''S'' → ''N''}} , {{math|''S'' ⇒ ''N''}} and "''N'' whenever ''S''".<ref>{{citation|first=Keith|last=Devlin|title=Sets, Functions and Logic / An Introduction to Abstract Mathematics|edition=3rd|publisher=Chapman & Hall|year=2004|isbn=978-1-58488-449-1|pages=22–23}}</ref>

In the above situation of "N whenever S," ''N'' is said to be a '''necessary''' condition for ''S''. In common language, this is equivalent to 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). In other words, the antecedent ''S'' cannot be true without ''N'' being true. For example, in order for someone to be called '''''S'''''ocrates, it is necessary for that someone to be '''''N'''''amed. Similarly, in order for human beings to live, it is necessary that they have air.<ref name=":2">{{Cite web|url=https://www.sfu.ca/~swartz/conditions1.htm#section3|title=The Concept of Necessary Conditions and Sufficient Conditions|website=www.sfu.ca|access-date=2019-12-02}}</ref>

One can also say ''S'' is a '''sufficient''' condition for ''N'' (refer again to 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, "the truth of ''S'' guarantees the truth of ''N''".<ref name=":2" /> For example, carrying on from the previous example, one can say that knowing that someone is called '''''S'''''ocrates is sufficient to know that someone has a '''''N'''''ame.

A '''''necessary and sufficient''''' condition requires that both of the implications <math>S \Rightarrow N</math> and <math>N \Rightarrow S</math> (the latter of which can also be written as <math>S \Leftarrow N</math>) hold. The first implication suggests that ''S'' is a sufficient condition for ''N'', while the second implication suggests 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 <math>S \Leftrightarrow N</math>.

{| class="wikitable" style="margin:1em auto; text-align:center;"
|+ Truth table
|-
! scope="col"  style="width:20%" | {{nobold|{{mvar|S}}}}
! scope="col"  style="width:20%" | {{nobold|{{mvar|N}}}}
! scope="col"  style="width:20%" | <math>S \Rightarrow N</math>
! scope="col"  style="width:20%" | <math>S \Leftarrow N</math>
! scope="col"  style="width:20%" | <math>S \Leftrightarrow N</math>
|-
| T || T || T || T || T
|-
| T || style="background:papayawhip" | F || style="background:papayawhip" | F || T || style="background:papayawhip" | F
|-
| style="background:papayawhip" | F || T || T || style="background:papayawhip" | F || style="background:papayawhip" | F
|-
| style="background:papayawhip" | F || style="background:papayawhip" | F || T || T || T
|}