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 :

semantics :

$[[x]]_I$
$[[y]]_I$
$[[x \Leftrightarrow y]]_I$
\(1\)
\(1\)
\(1\)
\(0\)
\(1\)
\(0\)
\(1\)
\(0\)
\(0\)
\(0\)
\(0\)
\(1\)

Notes
The equivalence of two propositions is only true if both propositions have the same truth value assigned.
Propositions, the equivalence of which is true, are called equivalent statements . Equivalent statements can be interpreted as the way of “saying the same” in two different ways.

| | | | | created: 2015-06-06 21:26:45 | modified: 2018-02-20 00:42:30 | by: bookofproofs

1. Corollary : Commutativity of Equivalence