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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
CS103 Today
• What we will study and why
Mathematical Foundations of Computing
• How the course will be run
Robert Plummer • Propositional logic
9/21/09
CS103
8
Mathematical Foundations of Computing Number of Lectures
7
6
5
Logic 4
3
2
Induction 1
0
Sets, Relations, and Functions
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Automata and Formal Languages
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Computability Theory Co
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Complexity Theory
Why Is CS103 Important?
Proof skills
16
15
14 Way of Thinking
13
12
Number of Lectures
11
10
Foundations of CS
9
8
7 Applications
6
5
Programming Languages
4 Compilers
3
2 Circuit Design
1
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AI
Foundation Theory Cryptography
Intellectually and Philosophically Interesting
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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
Course Details Online Course Support
Instructor: Bob Plummer
Gates 178 This quarter we will be using CourseWare, a new
Office hours M, W 3:30 – 5:00 course management system. Register for CS103 at
or by appointment
https://courseware.stanford.edu
TAs: Louis Chatriot
Jeremy Hiatt We also have an email hotline that is closely monitored
Tyler Mullen by course staff:
Ilana Segall
cs103staff@gmail.com
"Working Office Hours" to be announced
Textbooks Exams
1. Custom book for Logic portion of the class
Midterm Tuesday, Oct. 27, 7 – 9 pm
Introduction to Theory of Computation, CS103.
Instructor: Robert Plummer. Final Monday, Dec. 7, 12:15 – 3:15 pm
Available at the Bookstore only Locations to be announced
2. Text for the main part of the course
Alternate exams are given on an individual basis and
Sipser, Introduction to the Theory of Computation, 2nd Ed. and only for emergencies or official university business
Some course notes will also be distributed
Homework Policy Grading
9 written problem sets, counted equally
Due in class Midterm 20%
3 free late days (max. 1 on any assignment) Final 35%
Not accepted more than one class period late Homework 45%
Late without a late day: -10%
You may work with others in the class, but you
must write up your problem sets individually
Revealing an answer denies the other person
the opportunity for the "Aha!" experience
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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
The Flu The Flu
Course policies will be adjusted if necessary
A large outbreak will be a serious matter
Under consideration:
-- This class is available online;
-- More late days so if you have flu symptoms,
-- Alternate method of submission DON'T COME
-- Drop one assignment
-- Alternate exams
-- Take-home exams
-- More generous curve
For now, the announced policies are in effect
Propositional Calculus Propositional Calculus
A proposition is a declarative statement that is The first thing we will study is how to combine
either true or false (but not both). propositions into larger propositions and how
to determine the truth value of the results.
Bob is teaching this class. Negation
3>2 It is not the case that Bob is teaching this class.
It is snowing outside.
It is not the case that it is snowing outside.
This program sorts any list of positive integers
into non-decreasing order.
Logical Operators a.k.a. Boolean Connectives
Propositional Calculus
Negation ¬p is true if and only if p is false.
So that we can discuss the rules of propositional ¬p is read "not p".
calculus in a compact way, we will use propositional
variables such as p, q, r, ... to stand for propositions.
Disjunction p ∨ q is true if and only if
p is true, q is true, or both are true.
Propositional variables have one of two possible values: p ∨ q is read "p or q".
true, which we denote by T, and
false, which we denote by F.
Conjunction p ∧ q is true if and only if
We will use logical operators to describe the ways that
both p and q are true.
we can build larger propositions from smaller ones.
p ∧ q is read "p and q".
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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
Logical Operators a.k.a. Boolean Connectives Logical Operators a.k.a. Boolean Connectives
Negation p ¬p ¬p is true if and only if p is false. Negation p ¬p ¬p is true if and only if p is false.
T F ¬p is read "not p". T F ¬p is read "not p".
F T F T
Disjunction p ∨ q is true if and only if Disjunction p q p∨q p ∨ q is true if and only if
p is true, q is true, or both are true. T T T p is true, q is true, or both are true.
p ∨ q is read "p or q". T F T p ∨ q is read "p or q".
F T T
F F F
Conjunction p ∧ q is true if and only if Conjunction p q p∧q p ∧ q is true if and only if
both p and q are true. T T T both p and q are true.
p ∧ q is read "p and q". T F F p ∧ q is read "p and q".
F T F
F F F
In a truth table, we are not concerned with what p and q mean.
Each stands for a proposition that can be true or false.
Logical Operators a.k.a. Boolean Connectives Logical Operators a.k.a. Boolean Connectives
p ⊕ q is true if and only if p ⊕ q is true if and only if
Exclusive Or p q p⊕q p is true or q is true, but not both. Exclusive Or p q p⊕q p is true or q is true, but not both.
T T F T T F
T F T I.e., p ⊕ q is true if and only if T F T I.e., p ⊕ q is true if and only if
F T T p and q have different truth values. F T T p and q have different truth values.
F F F F F F
p ⊕ q is read "p exclusive or q". p ⊕ q is read "p exclusive or q".
We will use parentheses to build larger propositions.
For example
p⊕q
is true exactly when
(p ∨ q) ∧ ¬(p ∧ q)
is true.
Evaluating a compound expression Evaluating a compound expression
1. Locate the main connective. 1. Locate the main connective.
This will be the last one you This will be the last one you
evaluate. evaluate.
p q (p ∨ q) ∧ ¬(p ∧ q) p q (p ∨ q) ∧ ¬(p ∧ q)
T T T T 2. Work from the "inside out",
T F T F placing the value of each
F T F T sub-expression under its
F F F F connective.
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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
Evaluating a compound expression Evaluating a compound expression
The first two This will be next.
1. Locate the main connective. 1. Locate the main connective.
to evaluate.
This will be the last one you This will be the last one you
evaluate. evaluate.
p q (p ∨ q) ∧ ¬(p ∧ q) p q (p ∨ q) ∧ ¬(p ∧ q)
T T 2. Work from the "inside out", T T 2. Work from the "inside out",
T F placing the value of each T F placing the value of each
F T sub-expression under its F T sub-expression under its
F F connective. F F connective.
The values
go here. 3. Some of your evaluations
will involve columns on the
right side of the table.
Evaluating a compound expression Evaluating a compound expression
1. Locate the main connective.
This will be the last one you p q (p ∨ q) ∧ ¬(p ∧ q) p ⊕ q
evaluate.
p q (p ∨ q) ∧ ¬(p ∧ q) T T T F F T F
T T 2. Work from the "inside out", T F T T T F T
T F placing the value of each F T T T T F T
F T sub-expression under its F F F F T F F
F F connective.
3. Some of your evaluations The joint truth table shows that the two
will involve columns on the
expressions are logically equivalent (i.e.,
right side of the table.
that they have the same truth conditions).
4. You are done when you have
placed values under the main Rosen shows another way to do this on
connective. page 10.
Our notation for logical equivalence
is the symbol ≡
(p ∨ q) ∧ ¬(p ∧ q) ≡ p ⊕ q
p q r p ∨ ¬ (q ∨ ¬ (r ∧ p))
Note that ≡ is an abbreviation for "is equivalent to". T T T T
T T F F
It is NOT a logical operator.
T F T T
T F F F
F T T F
F T F F
F F T F
F F F F
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CS103 HO#3 9/21/09
Introduction, policies, propositional logic
p q r p ∨ ¬ (q ∨ ¬ (r ∧ p)) p q r p ∨ ¬ (q ∨ ¬ (r ∧ p))
T T T F T T T T T F T
T T F T F T T F T T F
T F T F T T F T F F T
T F F T F T F F T T F
F T T T F F T T T T F
F T F T F F T F T T F
F F T T F F F T T T F
F F F T F F F F T T F
p q r p ∨ ¬ (q ∨ ¬ (r ∧ p)) p q r p ∨ ¬ (q ∨ ¬ (r ∧ p))
T T T F T F T T T T T F T F T
T T F F T T F T T F T F T T F
T F T T F F T T F T T T F F T
T F F F T T F T F F T F T T F
F T T F T T F F T T F F T T F
F T F F T T F F T F F F T T F
F F T F T T F F F T F F T T F
F F F F T T F F F F F F T T F
How many binary logical operators can there be?
p q ?
p q r p ∨ ¬ (q ∨ ¬ (r ∧ p))
T T T T F T F T T T
T T F T F T T F T F
T F T T T F F T F T
T F F T F T T F F F
F T T F F T T F
F T F F F T T F
F F T F F T T F
F F F F F T T F
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