CHAPTER 1
Logic and Proofs
Section 1.1
Propositions and Connectives
1.1 PROPOSITIONS AND CONNECTIVES
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.
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)
Example: Let P: 19 is composite, Q: 45 is a multiple of 3.
Truth tables
These are the truth tables of
PЛQ, PVQ and ~P
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:
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.
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.
Example: Give a useful denial of: x is a positive integer.
SSSS
Solution: see the truth table
‫؛‬
Example: f is incseasing and f is concave upward
P: f is inceasing, Q: f is concave upward.
What is ~(P v Q)?