# Logic

> *Definition*: a statement is a sentence that is either true or false, never both.

> *Definition* **- Logical operators**: let $A$ and $B$ be assertions. 
> * The assertion $A$ and $B$ ($A \land B$) is true, iff both $A$ and $B$ are true.
> * The assertion $A$ or $B$ ($A \lor B$) is true, iff at least one of $A$ and $B$ is true.
> * The negation of $A$ ($\neg A$) is true iff $A$ is false.

> *Definition* **- Implies**: if $A$ and $B$ are assertions then the assertion if $A$ then $B$ ($A \implies B$) is true iff
> * $A$ is true and $B$ is true,
> * $A$ is false and $B$ is true,
> * $A$ is false and $B$ is false.
> 
> This also works the opposite way, if $B$ then $A$ ($A \Longleftarrow B$)

> *Definition* **- If and only if**: if $A$ and $B$ are assertions then the assertion $A$ if and only if $B$ (A \iff B) is true iff
> * $(A \Longleftarrow B) \land (a \implies B)$.
>
> This leads to the following table.