1. Chapter 1
Logic and Proofs

2. Propositions and Connectives
Section 1.1
Propositions and Connectives

3. (1) 4+2=6 1.1 Propositions and Connectives (2) All people are Muslims.
Propositions will be denoted by P,Q,S,…. and so on, the value T or F is called the truth value of the proposition..
Examples:
(1) 4+2=6
(2) All people are Muslims.
(3) x+1=5.
(4) She lives in Gaza.

4. Definition: A propositional form is a compouned propositions formed from more that two propositional variables.
Example: [~P V (Q V R)] Л (R Л S) (P Л Q) V (Q V R)

5. Example: Let P: 19 is composite, Q: 45 is a multiple of 3.

6. Truth tables
JHJ
These are the truth tables of PЛQ, PVQ and ~P

7. Example: Find the truth table of (P Л Q) V ~R
Example: Translate the following sentence into symboles: then compare it with the table above:

8. Example: P V ~P is a tautology called (Excluded middle)
Example: P V ~P is a tautology called (Excluded middle). It is true whatever the truth value of P.
Example: Show that ( P ∨ Q) ∨ (∼P ∧∼Q) is a tautology.

9. Definition: A denial of a proposition P is any proposition equivalent to ∼P. Note that a proposition has only one negation, ~P, but it has many denials. Give a useful denial of: x is a positive integer. Example:

10. ssss
Solution: see the truth table

11. ؛

12. Example: f is incseasing and f is concave upward P: f is inceasing, Q: f is concave upward. What is ~(P v Q)?