Chapter 1. The Foundations: Logic and Proofs Sections at zyBooks: 1.1 ...
[Pages:31]Lectures 1-2
CSI30
Chapter 1. The Foundations: Logic and Proofs
Sections at zyBooks:
1.1 Propositions and Logical Operations 1.2 Compound Propositions 1.3 Conditional Statements
Propositional Logic
CSI30
proposition ? is a sentence that declares a fact, i.e. declarative statement, that is either true or false, but not both.
examples: "It is raining now" 1+6 = 10 Washington, D.C. is the capital of United States of America
Propositional Logic
CSI30
proposition ? is a sentence that declares a fact, i.e. declarative statement, that is either true or false, but not both.
examples: "It is raining now" 1+6 = 10 Washington, D.C. is the capital of United States of America
not propositions: "What time is it now?" x+5=10 "Please, stand clear of the closing doors."
Do you see the difference between the two sets of examples? Why the last three sentences are not propositions?
Propositional Logic
CSI30
proposition ? is a sentence that declares a fact, i.e. declarative statement, that is either true or false, but not both. we use letters to denote propositional variables (i.e. variables that represent propositions): p, q, r, s examples: p : 1+7=10 q : "It is sunny outside" If a proposition is true, its value can be denoted by T or True or 1 If a proposition is false, its value can be denoted by F or (bottom) or
False or 0 The area of logic that deals with propositions is called the propositional logic or propositional calculus.
It was first developed by the Greek philosopher Aristotle. Can we build/construct new propositions?
Propositional Logic
CSI30
New propositions can be constructed from existing propositions using logical operators, and are called compound propositions.
logical operators:
?
negation
conjunction
disjunction
?p p q p q
other denotations:
p, not p p and q p or q
meaning:
"not p" "p and q" "p or q"
example 1: Let proposition p stand for "I will go to a movie theater", then ?p means "I will not go to a movie theater."
Truth table for negation operation:
p ?p
T
F
F
T
or
p ?p
1
0
0
1
Propositional Logic
CSI30
New propositions can be constructed from existing propositions using logical operators, and are called compound propositions.
logical operators:
?
negation
conjunction
disjunction
?p p q p q
other denotations:
p, not p p and q p or q
meaning:
"not p" "p and q" "p or q"
example 2: Let proposition p stand for "It is raining" and q stand for "I want to go to a movie theater", then pq means "It is raining and I want to go to a movie theater."
Conjunction p q is true when both p and q are true. Truth table for p q:
p
q pq
p
q pq
T
T
T
1
1
1
T
F
F
1
0
0
F
T
F
or
0
1
0
F
F
F
0
0
0
Propositional Logic
CSI30
New propositions can be constructed from existing propositions using logical operators, and are called compound propositions.
logical operators:
?
negation
conjunction
disjunction
?p p q p q
other denotations:
p, not p p and q p or q
meaning:
"not p" "p and q" "p or q"
example 3: Let proposition p stand for "It is raining" and q stand for "I want to go to a movie theater", then pq means "It is raining or I want to go to a movie theater."
Disjunction p q is true when at least one of p and q is true. Truth table
for p q:
p
q pq
p
q pq
T
T
T
1
1
1
T
F
T
or
1
0
1
F
T
T
0
1
1
F
F
F
0
0
0
Propositional Logic
CSI30
more logical operators:
implication p q
" if p then q", "p implies q" see page 6 for more
+ exclusive or p + q
"either p or q"
Implication: p q
hypothesis, or antecedent, or premise
conclusion, or consequence
if p is true then q holds
Implication is also called conditional statement.
Truth table for the implication p q :
p
q pq
T
T
T
T
F
F
F
T
T
F
F
T
example: Let p: "the weather is good", and q: "we'll go to the beach".
Then pq stands for "If the weather is good we'll go to the beach"
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