## 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 
- $$\forall{x\in{E}} \text{ is } \exists{ x \in{E}}$$
-