> 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-3-5.md).

# Logic and proofs(3/5)

## [Predicate Logic](https://www.quora.com/What-is-the-difference-between-proposition-and-predicate-in-math)

1/5, 2/5 講的是proposition logic(命題邏輯)，這邊要講predicate logic(述詞邏輯)

主詞，述詞(謂詞)，單單只有述詞時是沒有真假的，在說一個特徵。把主詞設為一個變數。

## Example.1

Alice is a girl.

* "Alice" is subject
* "is a girl" is predicate

P(x) = "x is a girl"

* P is a proposition function
* P(x) is the value of the proposition P at x
* For example, P(Andy) = "Andy is a girl"
* Note that P(x) is not a proposition

## Example.2

Let Q(x, y, z) = "x + y = z"

Q(1, 2, 3) is true.

在陳述一個式子，x + y = z，但在沒有把proposition variable帶進去前他不是proposition

## Example.3

Let P(x) = "x > 3"

P(2) is false

p(5) is true

## Universal Quantifier

∀x P(x)

all x,

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTOPIo7QlBc4h63qpc%2F-LZTOSVG2caMIDNcNJOF%2F1550993162861.jpg?alt=media\&token=76d9015a-d84c-4fd2-939e-da7492e3e028)

所有， 所有名字開頭都是B

## Existential Quantifier

∃x P(x)

有一個x

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTP6FH2xdr75pXsIc-%2F-LZTPVx01t9ryAYAOGjY%2F1550993335589.jpg?alt=media\&token=bb6d5389-ef8a-4aaf-8e10-9c7d8e2ebcee)

## 都可以冠上Demorgan

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTPcNTN-XGC8sx1pWy%2F-LZTQ4-Id8rCdNgTnPoi%2F1550993590340.jpg?alt=media\&token=8a1cc5a0-f288-497a-8b43-8da704a024c5)

universal 也同如此。

## Practice

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTlmxmJRwLcQB4HIv0%2F-LZTlw6kVidS7ebfiUBz%2F1550999555153.jpg?alt=media\&token=c23fdc2e-2df9-4001-9d5f-d92130f61063)

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTnagj2l6T9fOxV7D9%2F-LZTnl8ogyk2f4oc53Tf%2F1551000052336.jpg?alt=media\&token=95070707-6f7b-40eb-a259-c5f5c367b9a6)

## **存在唯一**

∃!x

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTolD4rgyBYCbATkW8%2F-LZTopaFZBLyejTKzjCC%2F1551000332252.jpg?alt=media\&token=b214d491-867e-4d01-8548-026bbae9412a)

because 有正負根號2

* **量詞也符合交換律，也可以把集合拿進去，因為集合的範圍也是個邏輯陳述。**
* **量詞的優先序是最大的。不能用括號來改變他的優先序。**

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTpT6Ycnso1QUAqj64%2F-LZTqNNEIBUYNJkZt1nb%2F1551000674083.jpg?alt=media\&token=a82f0a4d-c10f-41c4-8bd2-93ee82268539)

* 怎麼去理解以下這張圖

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTuvmqKZqeCCZe7_Wx%2F-LZTv2F2UVKN8rDttJAD%2F1551001928430.jpg?alt=media\&token=21129749-bce7-49af-9e26-b9d9ef810ba8)

1. 像是9 \* 9 雙層迴圈 每一個都要對應到
2. 像是9 \* 9 雙層迴圈隨便一個有就好

![](https://692803005-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LZAnL0SzCN0zyF32Pbe%2F-LZTwojFmbZRIuLI0iuL%2F-LZU0ELKa7JFObWdVswp%2F1234.jpg?alt=media\&token=e5303647-a17f-454e-b174-5392e881af4a)

注意!!  5 , 6 反過來了，更正下