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:
$[[x]]_I$ | $[[y]]_I$ | $[[x \Leftrightarrow y]]_I$ |
---|---|---|
\(1\) | \(1\) | \(1\) |
\(0\) | \(1\) | \(0\) |
\(1\) | \(0\) | \(0\) |
\(0\) | \(0\) | \(1\) |
Corollaries: 1
Branches: 1
Chapters: 2
Corollaries: 3
Examples: 4
Lemmas: 5 6 7 8
Motivations: 9
Parts: 10
Proofs: 11 12 13 14 15 16 17
Propositions: 18