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:

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

We read the equivalence

“$x$ *if, and only if* $y$.”

It has the following truth table:

$[[x]]_I$ | $[[y]]_I$ | $[[x \Leftrightarrow y]]_I$ |
---|---|---|

\(1\) | \(1\) | \(1\) |

\(0\) | \(1\) | \(0\) |

\(1\) | \(0\) | \(0\) |

\(0\) | \(0\) | \(1\) |

### 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.

| | | | | Contributors: *bookofproofs*

## 1.**Corollary**: Commutativity of Equivalence

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