course outline book: discrete mathematics by k. p. bogart topics: sets and statements symbolic logic...
TRANSCRIPT
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Course Outline
Book: Discrete Mathematics by K. P. BogartTopics:
Sets and statementsSymbolic LogicRelations functionsMathematical InductionCounting TechniquesRecurrence relationsTreesGraphs
Grades: First: 25%Second 25% Final 50%*Note: The outline is subject to change
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Discrete Mathematics
Is the one we use to analyze discrete processes that are carried out in a step-by-step fashion.
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Algorithm
A list of step by step instructions for carrying out a process
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Chapter 1
Sets and Statements
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Statements
A declarative sentence can be true, false or ambiguous
A statement is an unambiguous declarative sentence that is either true or false
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Example
5 plus 7 is 12 5 plus 7 is 5 5 plus 7 is large Did you have coffee this morning?
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Sets
Set: an unambiguous description of a collection of objects
EX:
Set of outcomes for flipping a coin
S={H,T}
However, the list of outcomes might be:
HTTTHHH…….
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Sets
Members of a set are called elements– aA “a is an element of A”
“a is a member of A”– aA “a is not an element of A”
EX: Set of +ve integersS={x |x>0}3 S-5 S
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Sets
Universe of a statement is the set whose elements are discussed by the statement
EX:x multiplied by x is +veThe universe could be:- Set of +ve integers- Set of –ve integers- Set of all integersFlipping a coin-Universe: {H,T}
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Sets
Note: P, q, r, s are used to represent statements X, y, z, w are used to represent variables
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Compound Statements
Simple statements are represented by symbolsEX: P: x is a positive integer Compound statements are represented by symbols+ logical
connectivesLogical Connectives:
– Conjunction AND. Symbol ^ – Inclusive disjunction OR Symbol v– Exclusive disjunction OR Symbol (+)– Negation Symbol ¬– Implication Symbol
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Compound Statements
Example:-I will take calculas1 and I will take physics class.Represented as: p ^ q- I will have coffee or I will have teaRepresented as: p v q- Ali is at school or Ali is at homeRepresented as: p (+) q- p: x is greater than 2 ¬p: x is not greater than 2-George is at school and either Sue is at store or Sue is at home.P ^( q (+) r )*Note the use of parentheses ( see example 4 page 7).
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Truth sets
The set of all values of x that make a symbolic statement p(x) true is called the truth set of the proposition p.
(the set of all values in the universe that makes p true).
The symbolic statements p(x) & q(x) are equivalent if they have the same truth sets.
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Truth sets
EX:Universe: The result of flipping 2 coins
P: the result has one head q: the result has one tail
P and q are equivalent since they have the same truth sets.
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Fundamental Principle of Set Equality
To show that the sets T and S are equal, we may show that each element in T is an element in S and vice versa.
EX:Universe: 300 coin flipsP: the result has 2 H’sq: the result has 298 T’sShow that p and q are equivalent.
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Finite and infinite sets
Finite sets - Examples:
A = {1, 2, 3, 4} B = {x | x is an integer, 1 < x < 4} D = {dog, cat, horse} D = {dog, cat, horse}
Infinite sets- Examples:
Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…} Natural numbers N = {0, 1, 2, 3, …} S={x| x is a real number and 1 < x < 4} = [0, 4]
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Section 1.2: Sets
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Venn diagrams
A Venn diagram provides a graphic view of sets and their operations: union, intersection, difference and complements can be identified
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Set operations
Given two sets X and Y the following are operations that can be performed on them:– Union– Intersection– Complement– Difference
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Union
The union of X and Y is defined as the set A B = { x | x A or x B}
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Intersection
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 =
XY
xy
XY
X Y =
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Complement
The complement of a set Y contained in a universal set U is the set Yc = U – Y
YUYc
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Difference
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
X YX-y
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Properties of set operations
Theorem : 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
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Properties of set operations (2)
c) Distributive laws: A(BC) = (A B) (A C) A(BC) = (A B) (A C)
d) Identity laws: AU=A A = A
e) Complement laws: AAc = U AAc =
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Properties of set operations (3)
f) Idempotent laws:
AA = A AA = A
g) Bound laws:
AU = U A =
h) Absorption laws:
A(AB) = A A(AB) = A
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Properties of set operations (4)
i) Involution law: (Ac)c = A
j) 0/1 laws: c = U Uc =
k) De Morgan’s laws for sets:
(AB)c = AcBc
(AB)c = AcBc
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Demorgan’s Laws for sets
~(A B) = (~A) (~B)
-Proof: To be discussed in class
~(A B) = (~A) (~B)
-Proof: exercise
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Theorem
Let p and q be statements and let P and Q be their truth sets, then:
- P Q is the truth set of p^q (proof discussed in class)
- P Q is the truth set of pvq- ~P is the truth set of ¬p
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Example: Venn Diagrams
Show that P (Q R) = (P Q) (P R)
Using Venn diagrams
- See example 9 page 18
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Subsets
It is a relation between sets ( not operation) A set S is a subset of set T if each element in S is also an
element in T. Examples:
A = {3, 9}, B = {5, 9, 1, 3}, is A B ?
A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, is A B ?
A = {1, 2, 3}, B = {2, 3, 4}, is A B ?
Equality: X = Y if X Y and Y X
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Subsets using Venn diagrams
The ellipse is a subset of the circle
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Theorem
Let R and S be two sets then:
- R and S are subsets of R S- R S is a subset of both R and S- R S = S if and only if R S- R S=R if and only if R S
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Example
Prove that
R (S T) S (R T)
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The Empty Set
The empty set has no elements.
Also called null set or void set.
EX:
P is the truth set of p: x>0
Q is the truth set of q: x<0
The truth set of p^q = P Q= P and Q are disjoint sets
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Section 1.3
Determining the Truth of Symbolic Statements
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Truth tables
Truth tables are used to determine truth or falsity of compound statements
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Truth table of conjunction
Truth table of conjunction
p ^ q is true only when both p and q are true.
p q p ^ q
T T T
T F F
F T F
F F F
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Truth table of disjunction
p q is false only when both p and q are false
p q p v q
T T T
T F T
F T T
F F F
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Exclusive disjunction
p (+) q is true only when p is true and q is false, or p is false and q is true. Example: p = "John is programmer, q = “John is a lawyer" p (+) q = "Either John is a programmer or John is a lawyer"
p q p (+) q
T T F
T F T
F T T
F F F
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Negation
Negation of p: in symbols ¬p
¬ p is false when p is true, ¬ p is true when p is false Example: p = "John is a programmer" ¬ p = "It is not true that John is a programmer"
p ¬ p
T F
F T
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Truth tables
Examples:
Truth table for :- ¬pvq- (pvq) ^ ¬(p^q)
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Definition
2 statements are equivalent if their truth tables have the same final column
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Exercise
Use the truth tables to find out whether the following statements are equivalent:
- (p^q) v (p^r)- P^(qvr)
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Section 1.4
The Conditional Connectives
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Conditional propositions and logical equivalence
A conditional proposition is of the form “If p then q” In symbols: p q Example:
– p = " John is a programmer"– q = " Mary is a lawyer "– p q = “If John is a programmer then Mary is a
lawyer"
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Truth table of p q
p q is true when both p and q are true
or when p is false
p q p q
T T T
T F F
F T T
F F T
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P q is equivalent to ¬pvq
Recall: 2 statements are equivalent if their truth tables have the same final column
Exercise:Show that p q and ¬p v q are equivalent.
Note: it is important to represent the implication() and the exclusive OR(+) using other connectives (^,V, ¬), why??
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Example
Rewrite without arrows:
¬r ( s v (r ^ t))
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Example
Consider flipping a coin 3 times p is the statement “ the first flip comes up
heads” q is the statement “there are at least 2
heads”
Find the truth sets of p, q, pq
Answer: {TTT,TTH,THT,THH,HHH,HHT,HTH}
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Section 1.5
Boolean Algebra:
When we apply known laws about set operations to derive other ones algebraically, we say we are doing Boolean Algebra.
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Example: ( not required)
Use Boolean algebra to prove the unique inverse property. if x P= and x P = U then x= ~Px = x U (identity law) = x (P ~P) (inverse law) = (x P) (x ~P) (distributive law) = (x ~P) (given property) = (P ~P) (x ~P) (Inverse law) = (P x) ~P (distributive law) = U ~P (given property) = ~P (Identity law)
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Boolean Algebra for statements
A formula says that 2 truth sets are equal corresponds to a formula saying that 2 statements are equivalent ( so all set laws are translated directly into statement laws).
The statements about a universe satisfy the following rules: a) Associativity: (p V q) V r = p v (q v r) (p ^ q) ^ r = p ^ (q^ r)
b) Commutativity: p V q = q V p p ^ q = q ^ p
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Boolean Algebra for Statements
c) Distributive laws: p ^ (q v r) = (p ^ q) V (p ^ r) p V ( q ^ r) = (p V q) ^(p V r)
d) Identity laws: p^1=p pV0 = p
e) Complement laws: p V ¬p = 1 p ^ ¬p = 0f) Idempotent laws: p V p = p p ^ p = p
g) Bound laws: p V 1 = 1 p ^ 0 = 0
h) Absorption laws:p v ( p ^ q ) = p p ^ ( p v q) = p
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i) Double negation law: ¬ ¬p = p
j) De Morgan’s laws:
¬(p V q) = ¬ p ^ ¬ q
¬(p ^ q) = ¬ p V ¬q
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Final Example
Simplify:- (¬ ¬r) V (s V (r ^ t))
Answer : r V s
- (¬ (r ^ s) V (r V s)) ^ (¬ (r V s) V (r ^ s))
Answer: (¬r ^ ¬s ) V (r ^ s)