> For the complete documentation index, see [llms.txt](https://mvdd188.gitbook.io/cs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mvdd188.gitbook.io/cs/san/logic-and-proofs-2-5.md).

# Logic and proofs(2/5)

## A compound proposition is always true or false

* tautology(always true)
* contradiction(always false)

注意一定要複合陳述

Ex: 

(1) tautology

* p or \~p

(2) contradiction

* p and \~p

## Def: Logic Equivalence

two compound propositions p and q are called logical equivalent, 

if p <--> is a tautology

當p ≡ q， p = true, q = true;  p = false, q = false

則 p <--> q is a tautology

if and only if 的 truth table

## Demorgan' s Law

以truth table 驗證

## **結合率，交換率**

Ex:&#x20;

* p and (q or r) ≡ (p and q) or (p and r)

&#x20;       比較特別的是在數的四則運算，乘法對加法滿足分配律，

&#x20;        X(Y + Z) = XY + XZ，但反之

**而Logic operator 內 or 跟 and 地位是一樣的，這邊比較不一樣所以**

* p or (q and r) ≡ (p or q) and (p or r)

![](/files/-LZT4qWcPoQB3WPm8ZM2)

## 重要的要記的，最基本的

![](/files/-LZT5hg9yjFiuzua9ErJ)

![](/files/-LZT6SGgbZHvTfFySpAQ)

## Algebra operation

## Truth table 可以拿來證明， 相對的Algebra 的運算也可以拿來推演。

Show p ^ q --> p v q is tautology


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