Download Chapter 1: The Foundations: Logic and Proofs

Chapter 1: The Foundations:
Logic and Proofs
1.1 Propositional Logic
1.2 Propositional Equivalences
1.3 Predicates and Quantifiers
1.4 Nested Quantifiers
1.5 Rules of Inference
1.6 Introduction to Proofs
1.7 Proof Methods and Strategy
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1.1: Propositional Logic
Propositions: A proposition is a
declarative sentence (that is, a
sentence that declares a fact) that
is either true or false, but not
both.
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Example 1:
All the following declarative sentences
are propositions:
1. Washington D.C., is the capital of
the USA.
2. Toronto is the capital of Canada
3. 1+1=2.
4. 2+2=3.
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Example 2:
Consider the following sentences. Are
they propositions?
1. What time is it?
2. Read this carefully.
3. x+1=2.
4. x+y=z
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• We use letters to denote propositional
variables (or statement variables).
• T: the value of a proposition is true.
• F: the value of a proposition is false.
• The area of logic that deals with
propositions is called the propositional
calculus or propositional logic.
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Let p and q are propositions:
Definition 1: Negation (Not)
• Symbol: ¬
• Statement: “it is not the case that p”.
• Example:
P: I am going to town
¬P:
It is not the case that I am going to town;
I am not going to town;
I ain’t goin’.
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Definition 2: Conjunction (And)
• Symbol:
• The conjunction pq is true when both p and q are
true and is false otherwise.
• Example:
P - ‘I am going to town’
Q - ‘It is going to rain’
PQ: ‘I am going to town and it is going to rain.’
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Definition 3: Disjunction (Or)
• Symbol:
• The disjunction pq is false when both p and q are
false and is true otherwise.
• Example:
P - ‘I am going to town’
Q - ‘It is going to rain’
P  Q: ‘I am going to town or it is going to rain.’
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Definition 4: Exclusive OR
• Symbol:
• The exclusive or of p and q, denote pq, is true
when exactly one of p and q is true and is false
otherwise.
• Example:

P - ‘I am going to town’
Q - ‘It is going to rain’
P  Q: ‘Either I am going to town or it is going
to rain.’
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Definition 5: Implication
•
•
•
•
•
If…. Then….
Symbol:
The conditional statement pq is false when p is true
and q is false, and true
P is called the hypothesis and q is called the conclusion.
Example:
P - ‘I am going to town’
Q - ‘It is going to rain’
P Q: ‘If I am going to town then it is going to rain.’
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Equivalent Forms
If P, then Q
P implies Q
If P, Q
P only if Q
P is a sufficient condition for Q
Q if P
Q whenever P
Q is a necessary condition for P
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Note: The implication is false only when P is true and
Q is false!
• ‘If the moon is made of green cheese then I have
more money than Bill Gates’ (?)
• ‘If the moon is made of green cheese then I’m on
welfare’ (?)
• ‘If 1+1=3 then your grandma wears combat boots’
(?)
• ‘If I’m wealthy then the moon is not made of green
cheese.’ (?)
• ‘If I’m not wealthy then the moon is not made of
green cheese.’ (?)
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More terminology
•QP is the CONVERSE of P  Q
•¬ Q  ¬ P is the CONTRAPOSITIVE of P  Q
•¬ P ¬ Q is the inverse of P  Q
•Example:
Find the converse of the following statement:
R: ‘Raining tomorrow is a sufficient condition
for my not going to town.’
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Procedure
Step 1: Assign propositional variables to component
propositions
P: It will rain tomorrow
Q: I will not go to town
Step 2: Symbolize the assertion
R: P  Q
Step 3: Symbolize the converse
QP
Step 4: Convert the symbols back into words
‘If I don’t go to town then it will rain tomorrow’
• Homework: Find inverse and contrapositive of statements
above.
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Definition 6: Biconditional
•
•
•
•
•
‘if and only if’, ‘iff’
Symbol:
The biconditional statement pq is true when p and q
have the same truth value, and is false otherwise.
Biconditional statements are also called bi-implications.
Example:
P - ‘I am going to town’
Q - ‘It is going to rain’
P Q: ‘I am going to town if and only if it is going
to rain.’
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Translating English
•Breaking assertions into component propositions look for the logical operators!
•Example:
‘If I go to Harry’s or go to the country I will not go
shopping.’
P: I go to Harry’s
Q: I go to the country
R: I will go shopping
If......P......or.....Q.....then....not.....R
(P Q)  ¬ R
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Constructing a truth table
1.
2.
3.
4.
one column for each propositional variable
one for the compound proposition
count in binary
n propositional variables = 2n rows
• Construct the truth table for
(P  ¬ Q)  (PQ)
• HW: Construct the truth table for (P Q)  ¬ R
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• What is the real meaning of ¬ PQ ?
a) (¬ P) Q
b) ¬ (PQ)
• What is the real meaning of PQR ?
a) (PQ)R
b) P(QR)
• What is the real meaning of P  QR ?
a) (P  Q)R
b) P  (QR)
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Logic and Bit Operations
• Example 20
Find the bitwise OR, bitwise AND, and
bitwise XOR of the bit strings
01 1011 0110 and
11 0001 1101.
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Logic Puzzles
• Example 18:
There are two kind of inhabitants, knights,
who always tell the truth, and their
opposites, knaves, who always lie. You
encounter two people A and B. What
are A and B if A says “B is a knight” and
B says “The two of us are opposite type”?
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Terms
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•
•
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•
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Proposition
Negation
Conjection
Disjunction
Exclusive OR
Implication
• Inverse
• Converse
• Contrapositive
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