However, this statement’s converse “If a number is divisible by 2, then it is divisible by 4” is false. It is often used to conjoin two statements which are logically equivalent. But what, precisely, does this statement mean? Sometimes the biconditional in the statement of the phrase “if and only if” is shortened to simply “iff.” Thus the statement “P if and only if Q” becomes “P iff Q.” ⟺ The confusion of these two statement forms is known as a converse error. If it is raining outside, then I take my umbrella with me on my walk. ", ThoughtCo uses cookies to provide you with a great user experience. The phrase “if and only if” is used commonly enough in mathematical writing that it has its own abbreviation. By using ThoughtCo, you accept our. To understand “if and only if,” we must first know what is meant by a conditional statement. We break this biconditional statement into a conditional and its converse. For an example of the phrase “if and only if” that involves statistics, look no further than a fact concerning the sample standard deviation. Synonyms for if and only if include provided, only if, presuming, assuming, if, providing that, as long as, assuming that, given that and on condition that. She will not leave any apple uneaten, and she will not eat any other type of fruit. This page was last changed on 4 September 2020, at 03:16. ). Then we see that this statement means both of the following: If we are attempting to prove a biconditional, then most of the time we end up splitting it. [1] or In logic and mathematics, if and only if (sometimes abbreviated as iff) is a logical operator denoting a logical biconditional (often symbolized by ↔ or ). The phrase “if and only if” is used commonly enough in mathematical writing that it has its own abbreviation. Note that the truth table shown is also equivalent to the XNOR gate. [3][4] In which case, A can be thought of as the logical substitute of B (and vice versa).[5]. What Are the Converse, Contrapositive, and Inverse? That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit. This statement is obtained from the original by saying “if Q then P.” Suppose we start with the conditional “if it is raining outside, then I take my umbrella with me on my walk.” The converse of this statement is “if I take my umbrella with me on my walk, then it is raining outside.”. We only need to consider the converse here. Biconditional statements are related to conditions that are both necessary and sufficient. Only if they win two will the issue be resolved before the final afternoon - and it will be difficult for a tiring squad. From Simple English Wikipedia, the free encyclopedia, "Iff" redirects here. In this case only the first condition is true, so FALSE is returned. To form a conditional statement, we could say “if P then Q.”. Sometimes the biconditional in the statement of the phrase “if and only if” is shortened to simply “iff.” Thus the statement “P if and only if Q” becomes “P iff Q.”, Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra. This construction eliminates some redundancy. An "if and only if" statement is also called a necessary and sufficient condition. Find more similar words at wordhippo.com! Consider the statement “if today is Easter, then tomorrow is Monday.” Today being Easter is sufficient for tomorrow to be Monday, however, it is not necessary. In this case, we may form what is known as a biconditional statement. Some Uses of "if and only if" in Writing About Mathematics Theorems which have the form "P if and only Q" are much prized in mathematics. One part we prove is “if P then Q.” The other part of the proof we need is “if Q then P.”. IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. These are called the converse, inverse, and the contrapositive. It is often used to conjoin two statements which are logically equivalent. If all of the data values are identical, then the standard deviation is equal to zero. {\displaystyle \leftrightarrow } For other uses, see, Wikipedia:How to write Simple English pages, "The Definitive Glossary of Higher Mathematical Jargon: If and Only If", https://simple.wikipedia.org/w/index.php?title=If_and_only_if&oldid=7096233, Pages needing to be simplified from September 2020, Creative Commons Attribution/Share-Alike License. Today could be any Sunday other than Easter, and tomorrow would still be Monday. "Madison will eat the fruit if and only if it is an apple" is equivalent to saying that "Madison will eat the fruit if the fruit is an apple, and will eat no other fruit". The following are examples of this kind of statement: Three other statements are related to any conditional statement. ↔ =IF(OR(A4>0,B4<50),TRUE, FALSE) IF A4 (25) is greater than 0, OR B4 (75) is less than 50, then return TRUE, otherwise return FALSE. What Does If and Only If Mean in Mathematics? We only need to consider this example to realize that the original conditional is not logically the same as its converse. We only need to look at a number such as 6. Sometimes you can prove an "If, and Only If" assertion without explicitly dividng the proof into two parts. This makes our proof have two parts. This brings us to a biconditional statement, which is also known as an "if and only if" statement. Certain conditional statements also have converses that are true. While the original statement is true, its converse is not.

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