?? ?? Discrete Mathematics

1
?? ??Discrete Mathematics
Hyogon Kim ??? 303 hyogon_at_korea.ac.kr / 3204
2
?? ??

• ??
• Discrete Mathematics(7/e), R. Johnsonbaugh,
  Prentice Hall, 2008
• ?? ??
• 2 Midterms 30 (9? ?, 10? ?)
• 4 Quizzes 25 (23? ??, from Homework)
• 1 Final 25 alpha
• 6 HWs/PGs 1218
• Plagiarism policy
• ?? ?? widen.korea.ac.kr ? Courses ? ????

3
8/28 Ch 1
30 Ch 1 Hw 1 Ch 1, 2
9/4 Ch 2
6 Ch 2
11 Ch 3
13 Ch 3 Hw 2 Ch 3, 4
18 Ch 4 Quiz 1 Ch 1, 2
20 Ch 4
25 Ch 5
27 Ch 5 Hw 3 Ch 5, 6
10/2 Ch 6
4 Ch 6 Quiz 2 Ch 3, 4
9 Ch 7
11 Ch 7 Hw 4 Ch 7, 8
16 Wed Exam 1 Ch 16
18 Ch 8
23 Ch 8
25 Ch 9
4
30 Ch 9
11/1 Ch 9 Hw 5 Ch 9, 10
6 Ch 10 Quiz 3 Ch 7, 8
8 Ch 10
13 Ch 11
15 Ch 11 Hw 6 Ch 11, 12
20 Wed Exam 2 Ch 710
22 Ch 12
27 Ch 12
29
12/4 - Quiz 4 Ch 11, 12
6 - Final All chapters
5
Discrete Mathematics 7th
edition, 2009

• Chapter 1
• Sets and logic
• Chapter 2
• Proofs

6
1.1 Sets

• Set a collection of distinct unordered objects
• Members of a set are called elements
• How to determine a set
• Listing
• Example A 1,3,5,7
• Description
• Example B x x 2k 1, 0 lt k lt 3

7
Finite and infinite sets

• Finite sets
• Examples
• A 1, 2, 3, 4
• B x x is an integer, 1 lt x lt 4
• Infinite sets
• Examples
• Z integers , -3, -2, -1, 0, 1, 2, 3,
  
• S x x is a real number and 1 lt x lt 4 1,
  4

8
Some important sets

• The empty set ? has no elements.
• Also called null set or void set.
• Universal set the set of all elements about
  which we make assertions.
• Examples
• U all natural numbers
• U all real numbers
• U x x is a natural number and 1lt xlt10

9
Cardinality

• Cardinality of a set A (in symbols A) is the
  number of elements in A
• Examples
• If A 1, 2, 3 then A 3
• If B x x is a natural number and 1lt xlt 9
• then B 9
• Infinite cardinality
• Countable (e.g., natural numbers, integers)
• Uncountable (e.g., real numbers)

10
Subsets , Power set

• X is a subset of Y if every element of X is also
  contained in Y
• (in symbols X ? Y)
• Equality X Y if X ? Y and Y ? X
• X is a proper subset of Y if X ? Y but Y ? X
• Observation ? is a subset of every set
• The power set of X is the set of all subsets of
  X, in symbols P(X),
• i.e. P(X) A A ? X
• Example if X 1, 2, 3,
• then P(X) ?, 1, 2, 3, 1,2, 1,3,
  2,3, 1,2,3
• Theorem 2.1.4 If X n, then P(X) 2n.

11
Set operations

• Given two sets X and Y
• The union of X and Y is defined as the set
• X ? Y x x ? X or x ? Y
• The intersection of X and Y is defined as the set
• X ? Y x x ? X and x ? Y
• Two sets X and Y are disjoint if X ? Y ?
• The difference of two sets
• X Y x x ? X and x ? Y
• The difference is also called the relative
  complement of Y in X
• Symmetric difference
• X ? Y (X Y) ? (Y X)
• The complement of a set A contained in a
  universal set U is the set Ac U A
  

12
Venn diagrams

• A Venn diagram provides a graphic view of sets
• Set union, intersection, difference, symmetric
  difference and complements can be identified

U
A
B
13
Properties of set operations

• Theorem 2.1.10
• Let U be a universal set, and A, B and C subsets
  of U.
• The following properties hold
• a) Associativity (A ? B) ? C A ? (B ? C)
• (A ? B) ? C A ? (B
  ?C)
• b) Commutativity A ? B B ? A
• A ? B B ? A
• c) Distributive laws A?(B?C) (A?B)?(A?C)
• A?(B?C) (A?B)?(A?C)
• d) Identity laws A?UA A??
  A
• e) Complement laws A?Ac U
  A?Ac ?

14
Properties of set operations

• f) Idempotent laws A?A A A?A A
• g) Bound laws A?U U A?? ?
• h) Absorption laws A?(A?B) A A?(A?B) A
• i) Involution law (Ac)c A
• j) 0/1 laws ?c U Uc ?
• k) De Morgans laws for sets
• (A?B)c Ac?Bc (A?B)c Ac?Bc

15
1.2 Propositions

• Logic the study of correct reasoning
• Statements as a single datum having (binary)
  truth value
• Representing facts digitally
• Q what about statements that have degree of
  truth value?
• How can we manipulate them to derive new things?
• Use of logic
• In mathematics
• to prove theorems
• In computer science
• to prove that programs do what they are supposed
  to do
• AI / DB Theorem prover
• Software engineering Program correctness

16
Propositions

• A proposition is a statement or sentence that can
  be determined to be either true or false.
• Examples
• John is a programmer" is a proposition
• I wish I were wise is not a proposition (?)
• Well, you can still assign some value
• Computers dont really care

17
Connectives

• If p and q are propositions, new compound
  propositions can be formed by using connectives
• Most common connectives
• Conjunction AND. Symbol
• Inclusive disjunction OR Symbol v
• Exclusive disjunction OR Symbol v
• Negation Symbol
• Implication Symbol ?
• Double implication Symbol ?

18
Truth table

• Truth table of conjunction
• , and, ???

• Truth table of Negation
• , not, ??

p q p q
T T T
T F F
F T F
F F F
p p
T F
F T

• Truth table of (inclusive) disjunction
• v, or, ???

• Truth table of exclusive disjunction
• Either p or q (but not both), ?, exclusive or

p q p v q
T T T
T F T
F T T
F F F
p q p v q
T T F
T F T
F T T
F F F
19
More compound statements

• Let p, q, r be simple statements
• We can form other compound statements, such as
• (p?q)r
• p?(qr)
• (p)?(q)
• (p?q)(r)
• and many others
• Example
• truth table of (p?q)r

p q r (p ? q) r
T T T T
T T F F
T F T T
T F F F
F T T T
F T F F
F F T F
F F F F
20
1.3 Conditional propositions (????) andlogical
equivalence (??)

• A conditional proposition
• If p then q , "p only if q"
• In symbols p ? q
• Truth table
• p antecedent or hypothesis (??)
• sufficient condition (?? ??) for q
• q consequent or conclusion (??)
• necessary condition (????) for p

p q p ? q
T T T
T F F
F T T
F F T
21
Logical equivalence

• logically equivalent
• ??
• two truth tables are identical.
• converse
• ?
• converse of p ? q is q ? p

• contrapositive
• ??
• contrapositive of the p ? q is q ? p.
• logically equivalent

p q p ? q p ? q
T T T T
T F F F
F T T T
F F T T
p q p ? q q ? p
T T T T
T F F F
F T T T
F F T T
p q p ? q q ? p
T T T T
T F F T
F T T F
F F T T
22
Logical equivalence

• double implication
• ???
• p if and only if q
• p ? q
• logically equivalent to
• (p ? q)(q ? p)

• tautology
• ?? ??
• truth table contains only true values for every
  case
• contradiction
• ?? ??
• truth table contains only false values for every
  case

p q p ? p v q
T T T
T F T
F T T
F F T
p q p ? q (p ? q) (q ? p)
T T T T
T F F F
F T F F
F F T T
p p (p)
T F
F F
23
De Morgans laws for logic

• The following pairs of propositions are logically
  equivalent
• (p ? q) and (p)(q)
• (p q) and (p) ? (q)

24
1.4 Arguments and rules of inference

• Deductive reasoning the process of reaching a
  conclusion q from a sequence of propositions
  p1, p2, , pn.
• The propositions p1, p2, , pn are called
• premises (??) or hypothesis (??).
• The proposition q that is logically obtained
  through the process is called the conclusion.

25
Rules of inference

• 1. Law of detachment or modus ponens
• (mode that affirms)
• p ? q
• p
• Therefore, q
• 2. Modus tollens
• (mode that denies)
• p ? q
• q
• Therefore, p
• 3. Rule of Addition
• p
• Therefore, p ? q

• 4. Rule of simplification
• p q
• Therefore, p
• 5. Rule of conjunction
• p
• q
• Therefore, p q
• 6. Rule of hypothetical syllogism
• p ? q
• q ? r
• Therefore, p ? r
• 7. Rule of disjunctive syllogism
• p ? q
• p
• Therefore, q

26
Rules of inference for quantified statements

• 1. Universal instantiation
• ?x?D, P(x)
• d ? D
• Therefore P(d)
• 2. Universal generalization
• P(d) for any d ? D
• Therefore ?x, P(x)

• 3. Existential instantiation
• ? x ? D, P(x)
• Therefore P(d) for some d ?D
• 4. Existential generalization
• P(d) for some d ?D
• Therefore ? x, P(x)

27
1.5 Quantifiers (????)

• A propositional function P(x) is a statement
  involving a variable x
• For example P(x) 2x is an even integer
• Domain of a propositional function
• if x is an element of a set D, D is called the
  domain of P(x)
• For example, x is an element of the set of
  integers
• the domain D of P(x) must be defined
• (cf.) P(x) x is an even integer
• if D is a set of even integers
• if D is a set of odd integers

28
Universal quantifier

• universal quantifier
• for every
• ?x P(x) P(x) for every x in a domain D
• True if P(x) is true for every x ? D
• False if P(x) is not true for some x ? D
• Example
• Let P(n) be the propositional function n2 2n is
  an odd integer ?n ? D all integers
• P(n) is true only when n is an odd integer,
• false if n is an even integer.

29
Existential quantifier, Counter example

• existential quantifier
• for some
• ?x P(x) P(x) for some x in a domain D
• true if there exists an element x in the domain D
  for which P(x) is true.
• counter example
• ?x P(x) is false if ?x?D such that P(x) is false.
• The value x that makes P(x) false is called a
  counter example to the statement ?x P(x).
• Example
• P(x) "every x is a prime number", for every
  integer x.
• But if x 4 (an integer) this x is not a primer
  number. Then 4 is a counter example to P(x)
  being true.

30
Generalized De Morgans laws for Logic

• If P(x) is a propositional function,
• then each pair of propositions in a) and b)
  below
• have the same truth values
• a) (?x P(x)) and ?x P(x)
• "It is not true that for every x, P(x)
  holds" is equivalent to
• "There exists an x for which P(x) is not
  true
• b) (?x P(x)) and ?x P(x)
• "It is not true that there exists an x for
  which P(x) is true" is
• equivalent to "For all x, P(x) is not true"

31
1.6 Nested Quantifiers

• Nested quantifier multiple quantifier
• Example
• ?x?y P(x,y)
• For every x in D, there is y in D such that
  P(x,y) is true
• ?x?y (xlty)
• For every x, there exist y such that xlty. D
  integer
• ?x?y L(x,y)
• L(x,y) x loves y
• Everybody(x) loves somebody(y)
• E.g. False there is someone who loves nobody
• ?x?y P(x,y)
• For every x in D and for every y in D, P(x,y) is
  true.
• ?x?y ((xgt0) (ygt0))?(xy gt 0)), D real number
• If there is at least one x and at least y such
  that P(x,y) is false
• Like in ?x?y ((xgt0) (ylt0))?(xy ? 0)), D
  real number
• Counter example x1, y-1

32
Nested Quantifier

• Example
• ?x?y P(x,y)
• There is at least on x such that P(x,y) is true
  for every y in D
• ?x?y (x?y), D positive integer
• x 1
• ?x?y (x?y), D positive integer
• False for every x, there is at least one
  positive integer y (ex. yx1)
• ?x?y P(x,y)
• There is at least one x in D and at least one y
  in D such that P(x,y) is true.
• ?x?y ((xgt1) (ygt1) (xy6))
• ?x?y ((xgt1) (ygt1) (xy7))

33
Nested Quantifier

• Xa,b,c, Y1,2,3,4

• ?x?y P(x,y)

• ?x?y P(x,y)

• ?x?y P(x,y)

• ?x?y P(x,y)

34
2.1 Mathematical systems, direct proofs and
counterexamples

• A mathematical system consists of
• Undefined terms
• Definitions
• Axioms

35
Undefined terms, Definitions

• Undefined terms are the basic building blocks of
  a mathematical system. These are words that are
  accepted as starting concepts of a mathematical
  system.
• Example in Euclidean geometry we have undefined
  terms such as
• Point
• Line
• A definition (??) is a proposition constructed
  from undefined terms and previously accepted
  concepts in order to create a new concept.
• Example In Euclidean geometry the following are
  definitions
• Two triangles are congruent if their vertices can
  be paired so that the corresponding sides are
  equal and so are the corresponding angles.
• Two angles are supplementary if the sum of their
  measures is 180 degrees.

36
Axioms, Theorems

• An axiom (??) is a proposition accepted as true
  without proof within the mathematical system.
• There are many examples of axioms in mathematics
• Example In Euclidean geometry the following are
  axioms
• Given two distinct points, there is exactly one
  line that contains them.
• Given a line and a point not on the line, there
  is exactly one line through the point which is
  parallel to the line.
• A theorem (??) is a proposition of the form p ?
  q which must be shown to be true by a sequence of
  logical steps that assume that p is true, and use
  definitions, axioms and previously proven
  theorems.

37
Lemmas and corollaries

• A lemma (????) is a small theorem which is used
  to prove a bigger theorem.
• A corollary (????) is a theorem that can be
  proven to be a logical consequence of another
  theorem.
• Example from Euclidean geometry "If the three
  sides of a triangle have equal length, then its
  angles also have equal measure."

38
Types of proof

• A proof is a logical argument that consists of a
  series of steps using propositions in such a way
  that the truth of the theorem is established.
• Direct proof p ? q
• A direct method of attack that assumes the truth
  of proposition p, axioms and proven theorems so
  that the truth of proposition q is obtained.

39
Direct proof example

• If n is even, then n2 is even.
• Suppose n is even. 
• Then by definition of even there is an integer
  m for which n 2m. 
• If we square both sides, we get n2 4m2 22m2.
  
• 2m2 is an integer because m is an integer, so by
  definition of even, n is even.

40
2.2 More methods of proof

• Proof by contradiction
• Proof by contrapositive
• Proof by cases
• Proofs of equivalence
• Existence proofs

41
Indirect proof

• The method of proof by contradiction (a.k.a.
  indirect proof) of a theorem p ? q consists of
  the following steps
• Assume p is true but q is false
• (p ? q) ? p ? q
• 2. Using p, q, axioms, previously derived
  theorems and rules of inference, derive r (r),
  a contradiction
• In particular, r could be p
• 3. Conclude q cannot be false, hence p ? q
• The only difference is the negated conclusion in
  our assumptions
• Use when direct proof is difficult

42
Special case proof by contrapositive

• If rp in the proof by contradiction, it is
  called the proof by contrapositive
• Since in effect we have shown (q) ? (p), which
  is logically equivalent to p ? q
• Example Show for every n ? Z, if n2 is even,
  then n is even.
• Suppose n2 is even, but n is odd.
• Then there exists an integer k s.t. n 2k1.
• If we square both sides we obtain n2 4k2 4k
  1 2(2k2 2k)1.
• But the equation tells us n2 is odd, a
  contradiction.
• We have proved that for every n ? Z, if n2 is
  even, then n is even.

43
Proof by contradiction example

• Prove ?2 is irrational.
• Suppose ?2 is rational.
• Then there exist integers p and q such that ?2
  p/q.
• Assume the fraction p/q is in lowest terms so
  that p and q are not both even.
• Squaring both sides gives 2 p2/q2
• Multiplying by q2 gives 2q2 p2.
• It means p2 is even, then p is even (!).
• Therefore, there exists an integer k s.t. p 2k.
  
• Substituting into 2q2 p2 gives q2 2k2.
• Therefore, q is even.
• Thus both p and q are both even, contradicting
  our assumption.
• Therefore, ?2 is irrational.

44
Others

• Proof by cases
• Prove (p1 ? p2 ? p3 ? ? pn) ? q
• Instead prove (p1 ? q) ? ? (pn ? q)
• Also called exhaustive proof
• Use when the number of cases to prove is small
• Proof of equivalence
• Prove p ? q
• Prove p ? q and q ? p
• Existence proof
• Prove ?x P(x)
• Just find one x that satisfies above and that is
  it
• Sometimes it is not so easy to find that x

45
2.3 Resolution proofs

• Due to J. A. Robinson (1965)
• A clause is a compound statement with terms
  separated by or, and each term is a single
  variable or the negation of a single variable
• Example p ? q ? (r) is a clause
• (p ? q) ? r ? (s) is not a
  clause
• Hypotheses and conclusion are written as clauses
• If a hypothesis is not a clause, it must be
  replaced by an equivalent expression that is
  either a clause or the and of clauses
• Only one rule
• If (p ? q) and (p ? r) are both true then (q ?
  r) is true
• Can be checked by truth table but think
• If (p ? q) ? (p ? r), wouldnt (q ? r) follow?

46
Resolution example

• Prove a ? b, a ? c, c ? d ? b ? d
• Resolution procedure
• a ? b, a ? c ? b ? c
• c ? b, c ? d ? b ? d
• Special cases
• If p ? q and p ? q is true
• If p and p ? r are true, ?r is true

47
Resolution example -- Prolog

• (just click on likes.p and make queries
• Sam's likes and dislikes in food
• Considering the following will give some
  practice
• in thinking about backtracking.
• ?- likes(sam,dahl).
• ?- likes(sam,chop_suey).
• ?- likes(sam,pizza).
• ?- likes(sam,chips).
• ?- likes(sam,curry).
• likes(sam,Food) - indian(Food), mild(Food).
• likes(sam,Food) - chinese(Food).
• likes(sam,Food) - italian(Food).
• likes(sam,chips).
• indian(curry).
• indian(dahl).
• indian(tandoori).

48
2.4 Mathematical induction

• Useful for proving statements of the form ?n?A
  S(n)
• N is the set of positive integers or natural
  numbers,
• A is an infinite subset of N
• S(n) is a propositional function

49
Mathematical Induction

• Suppose we want to show that for each positive
  integer n the statement S(n) is either true or
  false.
• 1. Verify that S(1) is true.
• 2. Let n be an arbitrary positive integer.
• Let i be a positive integer such that i lt n.
• 3. Show that S(i) true implies that S(i1) is
  true,
• i.e. show S(i) ? S(i1).
• 4. Then conclude that S(n) is true for all
  positive integers n.

50
Mathematical induction terminology

• Basis step
• Verify that S(1) is true.
• Inductive step
• Assume S(i) is true.
• Prove S(i) ? S(i1).
• Conclusion
• Therefore S(n) is true for all positive integers
  n.

51
(No Transcript)