epicours/Mathématiques/Séminaire/Logics/Logics.md

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
     !

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
   • !
   • 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
   • !

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 !
   • Exemple
     • Moustache => Nice
   • Let's take R=[P=>Q] >P$$ Is a converse of R=[P=>Q] The contrapositive of R=[P=>Q] is [\daleth{Q} => \daleth{P}] The both are equivalence

5. 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. !

     Let's take R = P\land{Q} !

     aleth{(P\land{Q})}=\daleth{P}\land{\daleth{Q}}$$

     !

     !

     >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}}