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One of the most important connectives in logic is the equivalence which is a kind of both-sided implication.

## Definition: Equivalence

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:

$$x \Leftrightarrow y :=(x\Rightarrow y)\wedge (y\Rightarrow y).$$

“$x$ if, and only if $y$.”

It has the following truth table:

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

### 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