Math Logic

Math Logic
Logic: the branch of mathematics that is in charge of deviating new properties from the ones we already know through valid and congruent reasoning. The part of math will be study here is propositional logic.
Propositional logic: a proposition is a statement that can be determent to be either true or false, but not both.
Proposition can be related to each other through connectors.The most common connectors are:
Common connectors:
^ | “and” | conjunction |
˅ | “or” | Exclusive disjunction |
→ | “if…then..” | implication |
↔ | “if and only if” | Double implication |
⌐ | “not” | Negation |
Basic rules for connectors :
Connector | Will be … | Only when… |
^ | T | TT |
˅ | F | FF |
→ | F | TF |
↔ | T | TT or FF |

Para saber cuantos T o F pone en lacolumna de una letra se debe de hacer : 2el numero de letras .
Ejemplo: 7P ˅ q
| p | q | 7p | 7p | v | q |
  | T | T | F | F | T | T |
  | T | F | F | F | F | F |
  | F | T | T | T | T | T |
  | F | F | T | T | T | F |

Aquí tenemos dos tipos de letra ( p y q) entonces para saber la cantidad de T y F total que deben de llevar las letras se hace 22 = 4. ( En la primera letra serian2 true y 2 false, ya que es la mitad.). Para encontrar cuantos true lleva la segunda letra se debe de hacer : el numero de T en la primera letra (p) dividido dos. (Lo mismo se hace con los F). Ya dividido se sigue un patrón hasta llegar al numero pedido de letras (4).

Logical equivalence
Two compound propositions are logically equivalent if and only if their tables of truth are equal.
P1≅P2↔ TP1=T(P2)
Examples:
1. Verify that p˅p ≅p
p | p˅p |
T | T |
F | F |
They are equal asi que:
Since Tp˅p=Tp, p˅p ≅p

2. Verify that ((7p ˅ q) ≅7q ˅ p
p | q | 7p | 7q | 7p˅q | 7q˅p |
T | T | F | F | T | T |
T | F | F | T | F | T |
F | T | T | F | T | F |
F | F | T | T | T | T |No son equal asi que:
Since T7p˅q≠T7q˅p,
7p ˅q ≠7q ˅p

Validity of an Argument
Arguments: an argument (À) is a set of propositions À=P1,P2,P3…Pn-1,Pn where P1,P2,P3…Pn-1 are called premises, and Pn is called the conclusion.
À=P1P2P3⋮Pn-1∴Pn
A tautology is a table of valueswhere our final conclusion is always true no matter what combination we use to get it.
Example:
(p→q) | ^ | 7q | → | 7p |
T | F | F | T | F |
F | F | T | T | F |
T | F | F | T | T |
T | T | T | T | T |
As you can see, the answer is a tautology because it is completely true. When it is completely false it is called a contradiction.
Validity of an argument: an argument is valid if andonly if (P1 ^P2^P3^…^Pn-1)→Pn is a tautology.
The argument of the previous example is :À=1. p→q2. 7q∴3. 7p

Conditional Proof
When proving the validity of an argument À in which the conclusion is an implication. For example:
À=P1P2P3⋮Pn-1∴A→B
It is equivalent to prove another argument where the antecedent of the implication has been added to the premises.
À=P1P2P3⋮Pn-1A∴BExample:
Prove that À is valid. À=1. p˅q2. 7p→q Ä=1. p˅q2. 7p3. q

Logical inference
1. Modus ponen 1. p→q2. p∴3. q

2. Modus Tollens1. p→q2. 7q∴3. 7p

3. Double Denial 1. 7(7p)∴2. p

4. Addition 1. p∴2. p ˅ q1. p∴2. p^q

5. Modus Tollendo Ponens 1. p˅q2. 7q∴3. p

6. Transitiavity1. p→q2. q→r3. p→r

7. Disjunctive Sillogism1. p→q2. r→s3. p ˅ r∴4. q ˅ s

8. Disjunctive Simplification 1. p ˅ q2. p→r3. q→r∴4. r

9. Destructive dilemma 1. p→q2. r→s 3.7q ˅ 7s∴4. 7p ˅ 7r

10. Simplification 1. p^q∴2. p 1. p^q∴2. q

11. Transportation 1. p→q∴2. 7q→7p

9.1 Sequences and Series

Infinite...