71 lines
2.7 KiB
Markdown
71 lines
2.7 KiB
Markdown
## Proposition
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A proposition (or assertion) is a sentence that can be a math or anything else that can be true or false. Logic use connector to create new proposition
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**Exemple** :
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- P="The sky is red"
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- Q="The is is not red"
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- It's a negation : $$Q=\daleth{P}$$ and $$P=\daleth{Q}$$
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## Connectors
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1) Negation
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- Def: Let P a proposition : we call negation of P a proposition or the proposition that is true when P is false and false when P is true. We denote this proposition :
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$$\daleth{P}$$
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- **Exemple :**
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- P="the sky is red"
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- $$\daleth{P} = The sky is not red$$
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![[Pasted image 20230904093718.png]]
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2) Conjunction
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- Def : let P and Q two proposition. We call conjunction of P and Q denote $$P1Q$$ the proposition that is :
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- True when P and Q are both true
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- False if else
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- ![[Pasted image 20230904094230.png]]
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- Exemple:
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- P: Floor is green
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- Q: Wall is withe
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- $$P\land{Q} = Floor is Green and the wall is white$$
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3) Disjunction
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- Def: Let P and Q a proposition. We call disjunction of P and Q to a proposition that is
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- True what at least one of $$P\land{Q}$$ is true and False when their both P and Q are false
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- ![[Pasted image 20230904095003.png]]
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4) If then (implication)
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- Def: Let P and Q two propositions. We call if P then Q the proposition that is :
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- False when P is true and Q is false. We denote $$P=>Q$$
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- True if else
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![[Pasted image 20230904100125.png]]
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- Exemple
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- Moustache => Nice
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- Let's take $$R=[P=>Q]$$
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$$Q=>P$$
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Is a converse of $$R=[P=>Q]$$
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The contrapositive of $$R=[P=>Q]$$ is $$[\daleth{Q} => \daleth{P}]$$
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The both are equivalence
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6) Equivalence
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- Def: Let P and Q two propositions. We say P and Q are equivalent and denote $$P=>Q$$ when both $$(P=>Q) and (Q=>P) are true$$ In other words when $$(P=>Q) and (Q=>P) is true$$
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and false if else.
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![[Pasted image 20230904100934.png]]
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Let's take $$R = P\land{Q}$$
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![[Pasted image 20230904101446.png]]
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$$\daleth{(P\land{Q})}=\daleth{P}\land{\daleth{Q}}$$
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![[Pasted image 20230904101721.png]]
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![[Pasted image 20230904102817.png]]
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$$A=>B$$
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A is the sufficient condition (to B) and B is the necessary condition (to A)
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$$A<=>B$$ A & B are the necessary and sufficient conditons
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## Quantifiers
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We know two quantifiers : $$\forall \text{ and } \exists$$
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**Example:** $$\forall{x \in{R}}, P(x)$$ is true whatever the values of x from R
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$$x^2\eqslantgtr{0}$$
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$$\exists{x\in{R}}, P(x)$$
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P is true for at lease one value of R
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$$\exists{x\in{R}}, x^2=0$$
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$$\exists{x\in{R}}, x^2=1$$
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### ⚠️ **The order maters for every quantifiers**
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- The negation of $$\exists{ x \in{E}} \text{ is } \forall{x\in{E}} $$
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- The negation of
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- $$\forall{x\in{E}} \text{ is } \exists{ x \in{E}}$$
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- |