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Mathématiques/Séminaire/Logics/Logics.md

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