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.

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

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