One of the most important connectives in logic is the equivalence which is a kind of both-sided implication.
Let \(L\) be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. An equivalence “\(\Leftrightarrow\)” is a Boolean function
\[\Leftrightarrow:=\begin{cases}L \times L & \mapsto L \\
(x,y) &\mapsto x \Leftrightarrow y\\
\end{cases}\]
defined using a conjunction of two implications:
$$x \Leftrightarrow y :=(x\Rightarrow y)\wedge (y\Rightarrow y).$$
We read the equivalence
“$x$ if, and only if $y$.”
It has the following truth table:
$[[x]]_I$ | $[[y]]_I$ | $[[x \Leftrightarrow y]]_I$ |
---|---|---|
\(1\) | \(1\) | \(1\) |
\(0\) | \(1\) | \(0\) |
\(1\) | \(0\) | \(0\) |
\(0\) | \(0\) | \(1\) |
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| created: 2015-06-06 21:26:45 | modified: 2018-02-20 00:42:30 | by: bookofproofs