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Definition (If And Only If $\iff$)

Let $P$ and $Q$ be statements or open sentences. We'll often encounter statements such as $P$ if and only $Q$ (for an example, see the next problem). The sentence $P$ if and only if $Q$, often written as $P$ iff $Q$ or as $P\iff Q$, is the conjunction $(P\implies Q)\wedge (Q\implies P).$

Note that $Q\implies P$ is the sentence "$P$ if $Q$" and note that $P\implies Q$ is the sentence "$P$ only if $Q$. So when we write $P$ if and only if $Q$, we are just saying ($P$ if $Q$) and ($Q$ if $P$) which is the same as saying ($Q\implies P) \wedge (P\implies Q)$.

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Page last modified on January 28, 2015, at 01:42 PM

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HomePage Schedule Pictures Writing Tips LaTeX Commands External Resources List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Definition If And Only If $\iff$ Please Login to access more options. Definition (If And Only If $\iff$) Let $P$ and $Q$ be statements or open sentences. We'll often encounter statements such as $P$ if and only $Q$ (for an example, see the next problem). The sentence $P$ if and only if $Q$, often written as $P$ iff $Q$ or as $P\iff Q$, is the conjunction $(P\implies Q)\wedge (Q\implies P).$ Note that $Q\implies P$ is the sentence "$P$ if $Q$" and note that $P\implies Q$ is the sentence "$P$ only if $Q$. So when we write $P$ if and only if $Q$, we are just saying ($P$ if $Q$) and ($Q$ if $P$) which is the same as saying ($Q\implies P) \wedge (P\implies Q)$. Edit Page History Source Attach File Backlinks List Group Page last modified on January 28, 2015, at 01:42 PM
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