Discrete Structures
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Lecture 1
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Administrative
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Syllabus
Lecture section: MWF 10-11
Lab/discussion section
Every week (mostly):
– Worksheet
– Quiz
– Homework
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Class webpage
Three midterms, as noted on the syllabus (on the class
webpage)
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Academic integrity
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Motivation
Why learn this material?
 Some things can be directly applied
 Some things are good to know
 Some things just teach a way of thinking and
expressing yourself
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Examples
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Directly applied- circuits to do addition
Conditionals in if statements
Things which are good to know- for example that the
square root of 2 can’t be stored
Way of thinking- the number of primes is infinite
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Overall theme
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Proofs of theorems
Examples of theorems
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Course topics
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Propositional logic (and circuits)
Predicate calculus - quantification
Elementary number theory
Mathematical induction
Sets
Counting - combinations and probability
Functions
Relations
Graph theory
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Chapter 1, Propositional Logic
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Statement/proposition
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Declarative (true or false)
Symbolized by a letter
Examples:
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–
–
–
–
–
–
–
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Today is Monday.
5+2=7
3 * 6 > 18
The sky is blue.
Why is the sky blue?
Barack Obama.
Two students in the class have a GPA of 3.275.
This sentence is false.
The current king of France is bald.
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Other symbols and definitions
for constructing compound statements
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Conjunction
"and" is symbolized by 
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Disjunction
"or" is symbolized by 
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Negation
"not" is symbolized by ~ (or sometimes )
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Truth table for 
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p
q
pq
1
1
1
1
0
0
0
1
0
0
0
0
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Truth table for 
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p
q
pq
1
1
1
1
0
1
0
1
1
0
0
0
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Truth table for 
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p
p
1
0
0
1
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Translation of English to symbolic logic
statements
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The sky is blue.
– one simple (atomic) statement- assign a letter or name to it, i.e., b
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The sky is blue and the grass is green.
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–
–
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one statement
conjunction of two atomic statements
each single statement gets a letter or name, i.e., b, g
and join with ^ i.e., b ^ g
The sky is blue or the sky is purple.
–
–
–
–
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one statement
disjunction of two atomic statements
each single statement gets a name, i.e., b, p
and join with  i.e., b  p
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Translation #1
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The sky is blue or purple.
– two statements
• "the sky is blue"- name this b
• "the sky is purple"- name this p
– it's still a disjunction
• "the sky is blue" or "the sky is purple"
• bp
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Translation #2
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The sky is blue but not dark.
– two statements
• "the sky is blue"- name this b
• "the sky is dark"- name this d
– conjunction with negation
• "the sky is blue" and "the sky is not dark"
• the sky is blue and it is not the case that the sky is dark
• "it is not the case that the sky is dark" is ~d
• b^~d
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Translation #3
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2x6
English: x is greater than or equal to 2 and less than
or equal to 6
Two statements:
– "x is greater than or equal to 2"- name this p
– "x is less than or equal to 6"- name this q
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Becomes p ^ q
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#3 continued- 2  x  6
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p is actually a compound statement
– x is greater than 2 or x is equal to 2
rs
– "x is greater than 2" is symbolized by r
– "x is equal to 2" is symbolized by s
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q is actually a compound statement
– x is less than 6 or x is equal to 6
– "x is less than 6" is symbolized by m
– "x is equal to 6" is symbolized by n
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mn
p ^ q becomes (r  s) ^ (m  n)
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The exclusive or
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The exclusive or of p and q is p or q, but not both
It’s indicated using 
It’s the same as (p  q)  (p  q)
p
q
pq
1
1
0
1
0
1
0
1
1
0
0
0
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Precedence between the operators
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~ (not)
has highest precedence
^ (and) /  (or) have equal precedence
use parentheses to override the operators' default
precedence, or when a statement has operators with
the same precedence, or just for clarity
a^bc
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Other "more advanced" truth tables
(p ^ r)  (q ^ r)
(p ^ ~r)  (p ^ r)
(p ^ q)  (~q  ~p)
(p ^ ~r)  (q  ~r)
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Discrete Structures
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Lecture 2
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Special truth tables
p  ~p
p ^ ~p
(p ^ ~q) vs. (~q ^ p)
(p ^ q) vs. ~ (~q  ~p)
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Special results in the truth table
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Tautological proposition
– a tautology is a statement that can never be false
– all of the lines of the truth table have the result "true"
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Contradictory proposition
– a contradiction is a statement that can never be true
– all of the lines of the truth table have the result "false"
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Logical equivalence of two propositions
– two statements are logically equivalent if they will be true in
exactly the same cases and false in exactly the same cases
– all of the lines of one column of the truth table have all of the
same truth values as the corresponding lines from another column
of the truth table
– it's indicated using 
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Logical equivalences
Theorem 1.1.1 - page 14
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Double negative:
~(~p)  p
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Commutative:
pqqp
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Associative:
(p  q)  r  p  (q  r)
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and p ^ q  q ^ p
and (p ^ q) ^ r  p ^ (q ^ r)
Distributive:
p ^ (q  r)  (p ^ q)  (p ^ r) and p  (q ^ r)  (p  q) ^ (p  r)
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Further logical equivalences
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Idempotent:
p^pp
Absorption:
p  (p ^ q)  p
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and p ^ ~p  c
Universal bound:
p^cc
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and p  c  p
Negation:
p  ~p  t
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and p ^ (p  q)  p
Identity:
p^tp
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and p  p  p
and p  t  t
Negations of t and c:
~t  c
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and ~c  t
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DeMorgan's laws
~(p  q)  ~p ^ ~q
– Example in English: It is not the case that Pete or Quincy went
to the store.  Pete did not go to the store and Quincy did not
go to the store.
~(p ^ q)  ~p  ~q
– Example in English: It is not the case that both Pete and Quincy
went to the store.  Pete did not go to the store or Quincy did
not go to the store.
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Prove by truth table and by rules
~(p  ~q)  (~q ^ ~p)  ~p
~((~p ^ q)  (~p ^ ~q))  (p ^ q)  p
(p  q) ^ ~(p ^ q)  (p ^ ~q)  (q ^ ~p)
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Discrete Structures
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Lecture 3
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Conditional statements
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Hypothesis  conclusion
If this, then that. The hypothesis implies the conclusion.
 has lowest precedence (~ / ^  / )
Examples in English:
– If it is raining, I will carry my umbrella.
– If you don’t eat your dinner, you will not get dessert.
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p
1
1
q
1
0
pq
1
0
0
0
1
0
1
1
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Converting  to 
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p  q  ~p  q
Show with truth table
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~(p  q)  p ^ ~q
Show with truth table and rules
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Contrapositive
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Given a conditional statement p  q, form its
contrapositive by:
– negating the conclusion and negating the hypothesis
– using the negated conclusion as the new hypothesis and the
negated hypothesis as the conclusion, e.g., ~q  ~p
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p  q  ~q  ~p
Examples in English:
– "If Paula is here, then Quincy is here."
– The contrapositive is "If Quincy is not here, then Paula is not here."
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Converse and inverse
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Given p  q, for example, "If Paula is here, then Quincy is
here."
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To form its converse:
– swap the hypothesis and the conclusion
– qp
– If Quincy is here, then Paula is here.
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To form its inverse:
– negate the hypothesis and negate the conclusion
– ~p  ~q
– If Paula is not here, then Quincy is not here.
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Only if
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Translation to if-then form
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p only if q
p can be true only if q is true
if q is not true then p can't be true
if not q then not p (~q  ~p)
if p then q (p  q)
Translation in English
– You will graduate is CS only if you pass this course.
• G only if P
– If you do not pass this course then you will not graduate in CS.
• ~P  ~G
– If you graduate in CS then you passed this course.
• GP
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Biconditional
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p if and only if q
pq
p iff q
p
q
pq
1
1
1
1
0
0
0
1
0
0
0
1
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Other English words for implication
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Sufficient condition
– "if r, then s" r  s
– means r is a sufficient condition for s
– the truth of r is sufficient to ensure the truth of s
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Necessary condition
– "if not x, then not y" ~x  ~y
– means x is a necessary condition for y
– the truth of x is necessary if y is true
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Sufficient and necessary condition
– p if, and only if q
pq
– the truth of p is enough to ensure the truth of q and vice versa
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Argument
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A sequence of statements where
– the last in the sequence is the conclusion
– all others are premises (assumptions, hypotheses)
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(premise1 ^ premise2 ^ …premisen)  conclusion
The critical rows of the truth table are those where all of
the premises are true
– only one premise being false makes the above conjunction false
– but a false hypothesis for a conditional can never make the
conditional's result be false
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The validity of an argument can be determined from the
truth value of the conclusion in the critical rows
– it's a valid argument if and only if the conclusion is true in all of
the critical rows
– it's an invalid argument if the conclusion is false in any of the
critical rows
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Discrete Structures
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Lecture 4
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Rules of Inference (Table 1.3.1- Page 39)
Modus Ponens
Modus Tollens
Elimination
pq
p
q
pq
~q
~p
pq
~q
p
Generalization
Specialization
Contradiction rule
p
pq
p^q
p
~p  c
p
q
pq
p^q
q
Transitivity
Conjunction
pq
qr
p r
p
q
p^q
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pq
~p
q
Proof by division into cases
pq
pr
qr
r
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Proofs using rules of inference
P1
P2
P3
pq
qr
~p
P1
p^q
P2
P3
ps
~r  ~q

r

s^r
P1 p  q
P2 ~(q  r)
P3 p  (m  r)
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~m
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Conditional worlds
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Making assumptions are only allowed if you go into a
"conditional world"
Two types of results can be derived when an assumption
is made: an implication and a negation
list of statements that are true in all worlds
assume anything
a list of statements which are true in the situations where the
assumption is true
conclusion: the assumption  anything from the
conditional world
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We'll call this "conditional world without contradiction"
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Conditional world assumption leads to
contradiction
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If you make an assumption, and that assumption leads to
a contradiction in the conditional world, what can be
concluded?
list of statements that are true in all worlds
assume anything
a list of statements which are true in the situations where the
assumption is true
a contradiction with something else known to be true
conclusion: the assumption must be false in all possible
worlds
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We'll call this "conditional world with contradiction"
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Prove using the conditional world methods
P1 (p  q)  s
P2 r  p
 rs
P1 m  s
P2 s  (q ^ r)
P3 q  ~r
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~(m ^ p)
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Discrete Structures
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Lecture 5
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Use both conditional world methods
P1
P2
P3
P4
~mp
p  (q  s)
~(s  ~x)
q  ~r

~(m ^ r)
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