csc331 week 1 topic a
TRANSCRIPT
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Topic ASets and Logic
Section 1.1Sets
Section 1.2Proposition
Section 1.3Conditional Propositions andLogical Equivalence
Section 1.4Arguments and Rules of
Inference
Section 1.5Quantifiers
Section 1.6Nested Quantifiers
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Section 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 < k < 3}
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Finite and infinite sets
Finitesets
Examples:
A = {1, 2, 3, 4}
B = {x | x is an integer, 1 < x < 4}
Infinitesets
Examples:
Z = {integers} = {, -3, -2, -1, 0, 1, 2, 3,} S={x| x is a real number and 1 < x < 4} = [0, 4]
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Some important sets
The emptyset has no elements.
Also called null setor void set.
Universalset: 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 1< x
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Some important sets
Two sets X and Y are equal and we write X = Y
if X and Y have the same elements.
X = Y if the following two conditions holds:
For every x, if x X, then x Y,
and
For every x, if x Y, then x X.
This is a great way to prove two sets are equal.
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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 1< x< 9}
then |B| = 9
Infinite cardinality Countable (e.g., natural numbers, integers)
Uncountable (e.g., real numbers)
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Subsets
X is a subsetof 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 aproper subsetof Y if X Y but
Y X Observation:is a subset of every set
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Set operations:
Union and Intersection
Given two sets X and Y
The unionof X and Y is defined as the set
X Y = { x | x X or x Y}
The intersectionof X and Y is defined as the set
X Y = { x | x X and x Y}
Two sets X and Y are disjointif X Y =
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Complement and Difference
The differenceof two sets
XY = { x | x X and x Y}
The difference is also called the relative complement
of Y in X
Symmetric difference
XY = (XY) (YX)
The complement of a set A contained in auniversal set U is the set Ac= UA
In symbols Ac = U - A
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Venn diagrams
A Venn diagram provides a graphic view ofsets
Set union, intersection, difference,
symmetric difference and complements canbe identified
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Properties of set operations (1)
Theorem 2.1.10: Let U be a universal set, and
A, B and C subsets of U. The followingproperties 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) = (AB)(AC)A(BC) = (AB)(AC)
d) Identity laws:
AU=A A= A
e) Complement laws:
AAc= U AAc=
Correction!!!!!It was incorrectly reversed in
previous version
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Properties of set operations (3)
f) Idempotent laws:
AA = A AA = Ag) 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 Morgans laws for sets:
(AB)c
= Ac
Bc
(AB)c= AcBc
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Union and intersection of
a family Sof sets
The unionof an arbitrary family Sof sets is
defined to be those elements x belonging to
at least one set X in S.
S = {x | x X for some X S}
The intersectionof an arbitrary family Sof
sets is defined to be those elements x
belonging to every set X in SS = {x | x X for all X S}
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Union and intersection of
a family Sof sets (2)
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Partition
Thepartitionof a set X divides X into non-
overlapping subsets.
More formally, a collection S of nonempty
subsets of X is said to be apartition of set X ifevery element in X belongs to exactly one
member of S
If S is a partition of X S is pair-wise disjoint and
S= X
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Logic
Logic = the study of correct reasoning
Use of logic In mathematics:
to prove theorems
In computer science:
to prove that programs do what they are
supposed to do
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Section 1.2 Propositions
Apropositionis a statement or sentence
that can be determined to be either true orfalse.
Examples:
John is a programmer" is a proposition I wish I were wise is not a proposition
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Truth table of conjunction
The truth values of compound propositionscan be described by truth tables.
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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Example
Let p = Tigers are wild animals
Let q = Chicago is the capital of Illinois p ^q = "Tigers are wild animals and
Chicago is the capital of Illinois"
p ^q is false. Why?
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Truth table of disjunction
The truth table of (inclusive) disjunctionis
p q is false only when both p and q are false Example: p = "John is a programmer", q = "Mary is a lawyer"
p v q = "John is a programmer or Mary is a lawyer"
p q p v q
T T T
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 isfalse 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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More compound statements
Let p, q, r be simple statements
We can form other compound statements,
such as (pq)^r
p(q^r)
(~p)(~q)
(pq)^(~r)
and many others
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Example: truth table of (pq)^r
p q r (p q)^rT T T
T T F
T F T
T F F
F T T
F T FF F T
F F F
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Section 1.3 Conditional propositions
and logical equivalence
A conditionalproposition 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 (true by default or vacuously true)
p q p q
T T T
T F F
F T T
F F T
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Hypothesis and conclusion
In a conditional proposition p q,
p is called the antecedentor hypothesis
q is called the consequentor conclusion
If "p then q" is considered logically the
same as "p only if q"
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Necessary and sufficient
A necessarycondition is expressed by the
conclusion.
A sufficientcondition is expressed by the
hypothesis. Example:
If John is a programmerthen Mary is a lawyer"
Necessary condition: Mary is a lawyer Sufficient condition: John is a programmer
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Logical equivalence
Two propositions are said to be logically
equivalentif their truth tables are identical.
Example: ~p q is logically equivalentto p q
p q ~p
q p q
T T T T
T F F F
F T T T
F F T T
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Converse
The converseof p q is q p
These two propositions
are not logically equivalent
p q p q q p
T T T T
T F F T
F T T F
F F T T
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Contrapositive
The contrapositive(or transposition) of the
proposition p q is ~q ~p.
They are logically equivalent.
p q p q ~q ~p
T T T T
T F F F
F T T T
F F T T
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Biconditional proposition
The biconditional propersitionp if and only if qis defined in symbols as p q
p q is logically equivalent to (p q)^(q p)
p q p q (p q) ^ (q p)
T T T T
T F F F
F T F F
F F T T
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De Morgans laws for logic
The following pairs of propositions are
logically equivalent:
~ (p q) and (~p)^(~q)
~ (p ^ q) and (~p) (~q)
S ti 1 4 A t d R l
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Section 1.4 Arguments and Rules
of Inference
Deductive reasoning: the process of reaching a
conclusion q from a sequence of propositions p1,
p2, , pn.
A (deductive) argument is a sequence ofpropositions written as
The symbol is read therefore.
The propositions p1, p2, , pnare calledpremisesor hypothesis.
The proposition q that is logically obtained
through the process is called the conclusion.
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Rules of inference (1)
1. Law ofdetachmentor
modus ponens p q
p
Therefore, q
2. Modus tollens
p q ~q
Therefore, ~p
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Rules of inference (2)
3. Rule ofAddition
p
Therefore, p q
4. Rule ofsimplification
p ^ q
Therefore, p
5. Rule of conjunction
p
q
Therefore, p ^ q
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Rules of inference (3)
6. Rule of hypothetical syllogism
p q
q r
Therefore, p r
7. Rule of disjunctive syllogism
p q
~p
Therefore, q
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Section 1.5 Quantifiers
Apropositional functionP(x) is a statement
involving a variable x
For example: P(x): 2x is an even integer
x is an element of a set D
For example, x is an element of the set of integers
D is called the domainof P(x)
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Domain of a propositional function
In the propositional function
P(x): 2x is an even integer,
the domain D of P(x) must be defined, for
instance D = {integers}.
D is the set where the x's come from.
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For everyand for some
Most statements in mathematics and
computer science use terms such as for
everyand for some.
For example:
For everytriangle T, the sum of the angles of T
is 180 degrees.
For everyinteger n, n is less than p, for someprime number p.
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Universal quantifier
One can write P(x) for everyx in a domain D
In symbols: x P(x)
is called the universal quantifier
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Truth of as propositional function
The statement x P(x) is
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.
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Existential quantifier
For somex D, P(x) is true if there exists
an element x in the domain D for which P(x) is
true. In symbols: x, P(x)
The symbol is called the existential
quantifier.
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Counterexample
The universal statement x P(x) is false ifx D such that P(x) is false.
The value x that makes P(x) false is called acounterexampleto 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 primernumber. Then 4 is a counterexample to P(x)being true.
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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 equivalentto "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) istrue" is equivalent to "For all x, P(x) is not true"
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Summary of propositional logic
In order to prove the
universally quantified
statement x P(x) is
true It is not enough to
show P(x) true for
some x D
You must show P(x) istrue for every x D
In order to prove the
universally quantified
statement x P(x) is
false It is enough to exhibit
some x D for which
P(x) is false
This x is called thecounterexampleto
the statement x P(x)
is true
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Rules of inference for
quantified statements
1. Universal instantiation
xD, 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)
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Section 1.6 Nested Quantifiers
By definition, the statement xyP(x, y),
with domain of discourse XY, is true if, for
everyxX and everyyY, P(x, y)is true.The statement xyP(x, y)is false if there is
at least onexX and at least oneyY
such that P(x, y)is false.
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xyP(x, y)
By definition, the statement xyP(x, y),with
domain of discourse XY, is true if, for every
xX, there is at least one yY for whichP(x, y)is true. The statement xyP(x, y)is
false if there is at least onexX such that
P(x, y)is falseforevery yY.
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xyP(x, y)
By definition, the statement xyP(x, y),with
domain of discourse XY, is true if there is at
least onexX such that P(x, y)is true forevery oneyY. The statement xyP(x, y)
is false if, for every xX, there is at least
oneyY such thatP(x, y)is false.
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xyP(x, y)
By definition, the statement xyP(x, y),with
domain of discourse XY, is true if there is at
least onexX and at least oneyY suchthat P(x, y)is true. The statement xyP(x,
y) is false if, for every xX and for every y
Y, P(x, y)is false.