01-Propositional_Logic

I Introduction¶

I.1 base notations¶

Our first building block is the notion of a proposition, which is simply a statement which is either true or false.

For example:

Notations we should know:

1. Conjunction（合取）: P∧Q (“P and Q”). True only when both P and Q are true.
2. Disjunction（析取）: P∨Q (“P or Q”). True when at least one of P and Q is true.
3. Negation（取反 / 否）: ¬P (“not P”). True when P is false.
4. Implication（蕴涵词）: P ⇒ Q (“P implies Q”). This is the same as “If P, then Q.”**
5. two-way implication p↔q

$$ \begin{aligned}&(\mathrm{a})\:\forall x\forall yP(x,y)\implies\forall y\forall xP(x,y).&&\text{True}\&(\mathrm{b})\:\forall x\exists yP(x,y)\implies\exists y\forall xP(x,y).&&\text{False}\&(\mathrm{c})\:\exists x\forall yP(x,y)\implies\forall y\exists xP(x,y).&&\text{True}\&(\mathrm{a})\:\forall x\:(P(x)\wedge Q(x))\stackrel{?}{\equiv}\:\forall x\:P(x)\wedge\forall x\:Q(x)\quad\textsf{T}\&(\mathrm{b})\:\forall x\:(P(x)\vee Q(x))\stackrel{?}{\equiv}\:\forall x\:P(x)\vee\forall x\:Q(x)\quad\textsf{F}\&(\mathrm{c})\:\exists x\:(P(x)\vee Q(x))\stackrel{?}{\equiv}\exists x\:P(x)\vee\exists x\:Q(x)\quad\textsf{T}\&(\mathrm{d})\:\exists x\:(P(x)\wedge Q(x))\stackrel{?}{\equiv}\exists x\:P(x)\wedge\exists x\:Q(x)\quad\textsf{F}\end{aligned} $$ (Detailed reason omission)

6. quantifiers: The universal quantifier ∀ (“for all”) and the existential quantifier ∃ (“there exists”).

We often write a proposition in the form of something like (∀x ∈ Z)(∃y ∈ Z)(x < y)

7. equivalent is something like:

• ¬(P∧Q) ≡ (¬P∨ ¬Q)
• ¬(P∨Q) ≡ (¬P∧ ¬Q)

Of course, these two formulas should be remembered since they tell us how to negate conjunctions and disjunctions

8. about P→Q, the truth table is shown below:(0 stands for F while 1 stands for T)

9. about P↔Q, the truth table is shown below:(0 stands for F while 1 stands for T)

10. We say that a sentence A entails another sentence B if in all models that A is true, B is as well, and we represent this relationship as A ⊨ B.

I.2 proposition formula¶

（穷举定理我们在 Proof by Cases（案例证明）中将会使用到）

I.3 logical equivalence¶

当命题 \(A\longleftrightarrow B\) 是重言式时，称 A 逻辑等价于 B，记作 \(A\equiv B\)。

实际上，符号 ⊨ 也是，但是打不出来，所以一般用 \(\equiv\) ；

逻辑等价：任何赋值情况下，A 和 B 都等值。

I.3.1 important logical equivalence¶

I.4 logical implication¶

当命题公式 A \(\to\) B 是重言式时，则称 A 逻辑蕴涵 B ，记作 A⊨B。

公式 A 的所有成真赋值都是公式 B 的成真赋值。

即任何赋值情况下，只要 A 为真，则 B 为真； \(A \equiv B\) 即为 \(A⊨B \land B⊨A\) 。

I.4.1 important logical implication¶

I.5 The important properties of logical equivalence and logical implication¶

I.6 ways to proof¶

I.7 priority of operations¶

1. 括号 ()：无论在哪个领域，括号始终具有最高的优先级，用于改变默认的优先级顺序。 2. 非 ~  !：在逻辑运算中，否定（逻辑非、位非）通常具有较高的优先级。 3. 与 ∧：这包括逻辑与（AND）、位与（&）。在没有括号改变顺序的情况下，它们通常在否定之后立即评估。 4. 异或 ⊕：在某些情况下，需要考虑异或运算（XOR），它可能在与运算和或运算之间。 5. 或 ∨：这包括逻辑或（OR）、位或（|）。它们在逻辑与之后进行评估。 6. 条件 →：如蕴含（→）通常优先级较低。 7. 双条件↔：双条件（↔）通常具有最低的优先级

II signs in latex¶

III Practice¶

[!QUOTE]

We need a lot of insight into propositions rather than just grasping concepts

For every real number k, there is a unique real solution to \(x^{3}\) = k.

(∀k ∈ R) (∃x ∈ R)(x 3 = k)∧(∀y,z ∈ R)(((y 3 = k)∧(z 3 = k)) ⇒ (y = z)) .

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