# Logic

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

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> *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.

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> *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 $A$ ($A \Longleftarrow B$)

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> *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.

| $A$   | $B$   | $A \implies B$ | $A \Longleftarrow B$ | $A \iff B$|
| :---: | :---: | :------------: | :------------------: | :-------: |
| true  | true  | true           | true                 | true      | 
| true  | false | false          | true                 | false     |
| false | true  | true           | false                | false     |
| false | false | true           | true                 | true      |

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> *Definition*: suppose $P$ and $Q$ are assertions. $P$ implies $Q$ if $P \implies Q$ is true. $P$ and $Q$ are equivalent if $P$ implies $Q$ and $Q$ implies $P$. 

## Methods of proof

*Direct proof*: for proving $P \implies Q$ only consider the case where $P$ is true.

*Proof by contraposition*: proving $P \implies Q$ to be true by showing that $\neg Q \implies \neg P$ is true.

*Proof by contradiction*: using the equivalence of $P \implies Q$ and $\neg Q \implies \neg P$ by assuming $P$ is not true and deducing a contradiction with some obviously true statement $Q$.

*Proof by cases*: dividing a proof into cases which makes use of the equivalence of $(P \lor Q) \implies R$ and $(P \implies R) \land (Q \implies R)$. Which together cover all situations under consideration.