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Updated syntax.

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Luc Bijl 2023-11-22 10:46:03 +01:00
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docs/en/mathematics/logic.md
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@ -1,6 +1,6 @@
# Logic # Logic
*Definition*: a statement is a sentence that is either true or false, never both. > *Definition*: a statement is a sentence that is either true or false, never both.
<br> <br>
@ -12,32 +12,32 @@
<br> <br>
*Definition* **- Implies**: if $A$ and $B$ are assertions then the assertion if $A$ then $B$ ($A \implies B$) is true iff > *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 true and $B$ is true,
* $A$ is false and $B$ is true, > * $A$ is false and $B$ is true,
* $A$ is false and $B$ is false. > * $A$ is false and $B$ is false.
>
This also works the opposite way, if $B$ then $A$ ($A \Longleftarrow B$) > This also works the opposite way, if $B$ then $A$ ($A \Longleftarrow B$)
<br> <br>
*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 > *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.
* $(A \Longleftarrow B) \land (a \implies B)$. | $A$ | $B$ | $A \implies B$ | $A \Longleftarrow B$ | $A \iff B$|
| :---: | :---: | :------------: | :------------------: | :-------: |
: This leads to the following table. | true | true | true | true | true |
| true | false | false | true | false |
| $A$ | $B$ | $A \implies B$ | $A \Longleftarrow B$ | $A \iff B$| | false | true | true | false | false |
| :---: | :---: | :------------: | :------------------: | :-------: | | false | false | true | true | true |
| true | true | true | true | true |
| true | false | false | true | false |
| false | true | true | false | false |
| false | false | true | true | true |
<br> <br>
*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 $P$. > *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 ## Methods of proof