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Version 10 |
| \section{Biconditional} |
\section{Biconditional} |
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A \emph{biconditional} is a truth function that is true only in the case that both parameters are true or both are false.
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A biconditional is a truth function that is true only in the case that both parameters are true or both are false.
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| Symbolically, the biconditional is written as |
Symbolically, the biconditional is written as |
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| $$ a \Leftrightarrow b$$ |
$$ a \Leftrightarrow b$$ |
| or |
or |
| $$ a \leftrightarrow b$$ |
$$ a \leftrightarrow b$$ |
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| with the latter being rare outside of formal logic. The truth table for the biconditional is |
with the latter being rare outside of formal logic. The truth table for the biconditional is |
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| \begin{center} |
\begin{center} |
| \begin{tabular}{ccc} |
\begin{tabular}{ccc} |
| a & b & $a \Leftrightarrow b$ \\ |
a & b & $a \Leftrightarrow b$ \\ |
| \hline |
\hline |
| F & F & T \\ |
F & F & T \\ |
| F & T & F \\ |
F & T & F \\ |
| T & F & F \\ |
T & F & F \\ |
| T & T & T |
T & T & T |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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| The biconditional function is often written as ``iff,'' meaning ``if and only if.'' |
The biconditional function is often written as ``iff,'' meaning ``if and only if.'' |
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| It gets its name from the fact that it is really two conditionals in conjunction, |
It gets its name from the fact that it is really two conditionals in conjunction, |
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| $$ (a \rightarrow b) \land (b \rightarrow a) $$ |
$$ (a \rightarrow b) \land (b \rightarrow a) $$ |
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| This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently. |
This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently. |
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| \section{Colloquial Usage} |
\section{Colloquial Usage} |
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The only unambiguous way of stating a biconditional in plain English is of the form ``$b$ if $a$ and $a$ if $b$.'' Slightly more formal, one would say ``$b$ implies $a$ and $a$ implies $b$.'' The plain English ``if'' may sometimes be used as a biconditional. One must weigh context heavily.
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The only unambiguous way of stating a biconditional in plain English is of the form ``$b$ if $a$ and $a$ if $b$.'' Slightly more formal, one would say ``$b$ implies $a$ and $a$ implies $b$.'' The plain english ``if'' may sometimes be used as a biconditional. One must weigh context heavily.
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| For example, ``I'll buy you an ice cream if you pass the exam'' is meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, ``it is cloudy if it is raining'' is \emph{not} meant as a biconditional, since it can obviously be cloudy while not raining. |
For example, ``I'll buy you an ice cream if you pass the exam'' is meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, ``it is cloudy if it is raining'' is \emph{not} meant as a biconditional, since it can obviously be cloudy while not raining. |
