In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It does not, however, exclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. This states that the only fruit Madison will eat is an apple. "Madison will eat the fruit only if it is an apple." (equivalent to " If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → the fruit is an apple").That the fruit is an apple is a sufficient condition for Madison to eat the fruit. All that is known for certain is that she will eat any and all apples that she happens upon. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. This states that Madison will eat fruits that are apples. "Madison will eat the fruit if it is an apple." (equivalent to " Only if Madison will eat the fruit, can it be an apple" or "Madison will eat the fruit ← the fruit is an apple").( June 2013) ( Learn how and when to remove this template message) Unsourced material may be challenged and removed. Please help improve this section by adding citations to reliable sources. The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention to interpret "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover"). However, this logically correct usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". Technically, definitions are "if and only if" statements some texts - such as Kelley's General Topology - follow the strict demands of logic, and use "if and only if" or iff in definitions of new terms. The authors of one discrete mathematics textbook suggest: "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as. However, in the preface of General Topology, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'". In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". It is somewhat unclear how "iff" was meant to be pronounced. Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'-but I could never believe I was really its first inventor." Usage of the abbreviation "iff" first appeared in print in John L. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts-that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false. Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". In logical formulae, logical symbols, such as ↔ via command \iff or \Longleftrightarrow. Some authors regard "iff" as unsuitable in formal writing others consider it a "borderline case" and tolerate its use. In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"-with its pre-existing meaning. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. then") combined with its reverse ("if") hence the name. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if. In logic and related fields such as mathematics and philosophy, " if and only if" (shortened as " iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
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