###### tags: `math`

# ==命題邏輯 propositional logic==
>aka語句邏輯 (sentential logic)
:::warning
Proposittion命題
>非對即錯 -> A true or false statement
:::

### <font color="#0F76D0 ">**Compound/complex Proposition 合句/複合句**</font>
:::info
優先規則：¬高於∧，∧高於∨，∨高於→，→高於↔
:::
#### 1. Negation非 ¬ (¬p)

| p | ¬p |
|:---:|:---:|
| T | F |
| F | T |

#### 2. Conjunction與 ∧ (p∧q)

#### 3. Disjunction或 ∨ (p∨q)

| p | q | p∧q | p∨q |
|:---:|:---:|:---:|:---:|
| T | T | T | T |
| T | F | F | T |
| F | F | F | F |

#### 4. Implication條件 → (p→q)
>if p, then q(p為假設，q為結果)

#### 5. Biconditional雙條件 ↔ (p↔q)
>p if and only if q(只要)

| p | q | p→q | p↔q |
|:---:|:---:|:---:|:---:|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |

### <font color="#0F76D0 ">**Conditional statements 條件陳述**</font>
>implycation : p->q
#### 1. converse逆命題 : q->p
#### 2. contrapositive逆否命題 : ¬q->¬p
#### 3. inverse否命題 : ¬p->¬q

### <font color="#0F76D0 ">**Equivalent propositions 等效命題**</font>
>Tow propositions are *equivalent* if they always have the same truth value

### <font color="#0F76D0 ">**Ｎon-equivalence 不等價**</font>
>用真值表(true table)表示命題（implication）的反轉和倒數都不等同於命題

# ==Application 應用==
### <font color="#0F76D0 ">**logic circuits 邏輯電路**</font>
* 0 represents **False**
* 1 represents **True**
* 如果變量的值爲true或false，則該變量稱爲布爾變量(Boolean variable)
* three basic circuits(gates)
- the inverter(the NOT Gate) -> negation
- the OR gate -> disjunction
- the AND gate -> conjunction

### <font color="#0F76D0 ">**consistent system specification**</font>
consistent : 有解的
> 有點像解方程式，帶入T或F，若句子皆成立便為 specifications consistent

# ==謂詞邏輯Predicate Logic==
>謂詞 -> 用來描述物與物之間關係的詞項
### <font color="#0F76D0 ">**Introducing**</font>
* variables(主體) : x, y, z
* predicates(描述) : P(x), M(x)
* quantifiers(範圍) : the extent to which a predicate is true
->**propositional function** P at x : P(x)

# ==Propositional Equivalences命題等價==
### <font color="#0F76D0 ">**Logical status 邏輯狀態**</font>
#### 1. tautology恆真句
>所有真值賦與下都為T : p∨¬p
#### 2. inconsistent/contradictory sentence不一致句
>所有真值賦與下都為F : p∧¬p
#### 3. contingent sentence適然句
>不為恆真句&&不一致句的剩下句子 : p

### <font color="#0F76D0 ">**Logical equivalence 邏輯等價**</font>
* logically equivalent if p↔q is a tautology(恆真句)
> write as p⇔q || p≡q
* compound proposition :

### <font color="#0F76D0 ">**De Morgan's laws 德摩根定律**</font>
* key logical equivalences
- **identity laws** : p∧T ≡ p, p∨F ≡ p
- **domination laws** : p∨T ≡ T, p∧F ≡ F
- **idempotent laws** : p∨p ≡ p, p∧p ≡ p
- **double negation laws** : ¬(¬p) ≡ p
- **negation laws** : p∨¬p ≡ T, p∧¬p ≡ F
* key logical equivalences(cont)
- **commutative laws** :
- **associative laws** :
- **distributive laws** :
- **absorption laws** :

### <font color="#0F76D0 ">**Propositional satisfiability 命題可滿足性**</font>