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'biconditional'
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| Title of object: |
biconditional |
| Canonical Name: |
Biconditional |
| Type: |
Definition |
| Created on: |
2001-10-24 20:34:28-04 |
| Modified on: |
2002-07-26 12:12:04.284884-04 |
| Classification: |
msc:03-00 |
| Synonyms: |
biconditional=iff |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
A biconditional is a truth function that is true only in the case that both parameters are true or both are false. For example, "a only if b", "a just in case b", as well as "b implies a and a implies b" are all ways of stating a biconditional in english. Symbolically the biconditional is written as
$$ a \leftrightarrow b$$
$$ a \Leftrightarrow b$$
Its truth table is
\begin{center}
\begin{tabular}{ccc}
a & b & $a \leftrightarrow b$ \\
\hline
F & F & T \\
F & T & F \\
T & F & F \\
T & T & T
\end{tabular}
\end{center}
In addition, the biconditional function is sometimes written as "iff", meaning "if and only if".
The biconditional gets its name from the fact that it is really two conditionals in conjunction,
$$ (a \rightarrow b) \land (b \rightarrow a) $$
This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently. |
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