course outline book: discrete mathematics by k. p. bogart topics: sets and statements symbolic logic...
TRANSCRIPT
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
Discrete Mathematics
Is the one we use to analyze discrete processes that are carried out in a step-by-step fashion.
Algorithm
A list of step by step instructions for carrying out a process
Chapter 1
Sets and Statements
Statements
A declarative sentence can be true, false or ambiguous
A statement is an unambiguous declarative sentence that is either true or false
Example
5 plus 7 is 12 5 plus 7 is 5 5 plus 7 is large Did you have coffee this morning?
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…….
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
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}
Sets
Note: P, q, r, s are used to represent statements X, y, z, w are used to represent variables
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
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).
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.
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.
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.
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]
Section 1.2: Sets
Venn diagrams
A Venn diagram provides a graphic view of sets and their operations: union, intersection, difference and complements can be identified
Set operations
Given two sets X and Y the following are operations that can be performed on them:– Union– Intersection– Complement– Difference
Union
The union of X and Y is defined as the set A B = { x | x A or x B}
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 =
Complement
The complement of a set Y contained in a universal set U is the set Yc = U – Y
YUYc
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
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
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 =
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
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
Demorgan’s Laws for sets
~(A B) = (~A) (~B)
-Proof: To be discussed in class
~(A B) = (~A) (~B)
-Proof: exercise
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
Example: Venn Diagrams
Show that P (Q R) = (P Q) (P R)
Using Venn diagrams
- See example 9 page 18
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
Subsets using Venn diagrams
The ellipse is a subset of the circle
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
Example
Prove that
R (S T) S (R T)
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
Section 1.3
Determining the Truth of Symbolic Statements
Truth tables
Truth tables are used to determine truth or falsity of compound statements
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
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
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
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
Truth tables
Examples:
Truth table for :- ¬pvq- (pvq) ^ ¬(p^q)
Definition
2 statements are equivalent if their truth tables have the same final column
Exercise
Use the truth tables to find out whether the following statements are equivalent:
- (p^q) v (p^r)- P^(qvr)
Section 1.4
The Conditional Connectives
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"
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
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??
Example
Rewrite without arrows:
¬r ( s v (r ^ t))
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}
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.
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)
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
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
i) Double negation law: ¬ ¬p = p
j) De Morgan’s laws:
¬(p V q) = ¬ p ^ ¬ q
¬(p ^ q) = ¬ p V ¬q
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)