Transcript Discrete Mathematics Lecture 4 Harper Langston New York University

Discrete Mathematics
Lecture 4
Harper Langston
New York University
Sequences
• Sequence is a set of (usually infinite number of)
ordered elements: a1, a2, …, an, …
• Each individual element ak is called a term, where
k is called an index
• Sequences can be computed using an explicit
formula: ak = k * (k + 1) for k > 1
• Alternate sign sequences
• Finding an explicit formula given initial terms of
the sequence: 1, -1/4, 1/9, -1/16, 1/25, -1/36, …
• Sequence is (most often) represented in a
computer program as a single-dimensional array
Sequence Operations
• Summation: , expanded form, limits
(lower, upper) of summation, dummy index
• Change of index inside summation
• Product:  , expanded form, limits (lower,
upper) of product, dummy index
• Factorial: n!, n! = n * (n – 1)!
Review Mathematical Induction
• Principle of Mathematical Induction:
Let P(n) be a predicate that is defined for
integers n and let a be some integer. If the
following two premises are true:
P(a) is a true
k  a, P(k)  P(k + 1)
then the following conclusion is true as well
P(n) is true for all n  a
Applications of Mathematical
Induction
• Show that 1 + 2 + … + n = n * (n + 1) / 2
(Prove on board)
• Sum of geometric series:
r0 + r1 + … + rn = (rn+1 – 1) / (r – 1)
(Prove on board)
Examples that Can be Proved
with Mathematical Induction
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Show that 22n – 1 is divisible by 3 (in book)
Show (on board) that for n > 2: 2n + 1 < 2n
Show that xn – yn is divisible by x – y
Show that n3 – n is divisible by 6 (similar to book
problem)
Strong Mathematical Induction
• Utilization of predicates P(a), P(a + 1), …, P(n)
to show P(n + 1).
• Variation of normal M.I., but basis may contain
several proofs and in assumption, truth assumed
for all values through from base to k.
• Examples:
• Any integer greater than 1 is divisible by a prime
• Existence and Uniqueness of binary integer
representation (Read in book)
Well-Ordering Principle
• Well-ordering principle for integers: a set
of integers that are bounded from below
(all elements are greater than a fixed
integer) contains a least element
• Example:
• Existence of quotient-remainder
representation of an integer n against
integer d
Stepping back: Algorithms
• Last lecture we talked about some elementary
number theory such as the Quotient Remainder
Theorem (pg 157)
• Algorithm is step-by-step method for performing
some action (such as finding remainder in Q.R.)
• Cost of statements execution
– Simple statements
– Conditional statements
– Iterative statements
Example: Division Algorithm
• Input: integers a and d
• Output: quotient q and remainder r
• Body:
r = a; q = 0;
while (r >= d)
r = r – d;
q = q + 1;
end while
Greatest Common Divisor
• The greatest common divisor of two
integers a and b is another integer d with
the following two properties:
– d | a and d | b
– if c | a and c | b, then c  d
• Lemma 1: gcd(r, 0) = r
• Lemma 2: if a = b * q + r, then gcd(a, b) =
gcd(b, r)
Euclidean Algorithm
• Input: integers a and b (a>b>=0)
• Output: greatest common divisor gcd
• Body:
r = b;
while (b > 0)
r = a mod b;
a = b;
b = r;
end while
gcd = a;
Exercise
• Least common multiple: lcm
• Prove that for all positive integers a and b,
gcd(a, b) = lcm(a, b) iff a = b
Correctness of Algorithms
• Assertions
– Pre-condition is a predicate describing initial state
before an algorithm is executed
– Post-condition is a predicate describing final state
after an algorithm is executed
• Loop guard
• Loop is defined as correct with respect to its preand post- conditions, if whenever the algorithm
variables satisfy the pre-conditions and the loop
is executed, then the algorithm satisfies the
post-conditions as well
Loop Invariant Theorem
• Let a while loop with guard G be given together with its
pre- and post- conditions. Let predicate I(n) describing
loop invariant be given. If the following 4 properties hold,
then the loop is correct:
– Basis Property: I(0) is true before the first iteration of the loop
– Inductive Property: If G and I(k) is true, then I(k + 1) is true
– Eventual Falsity of the Guard: After finite number of iterations, G
becomes false
– Correctness of the Post-condition: If N is the least number of
iterations after which G becomes false and I(N) is true, then
post-conditions are true as well
Correctness of Some Algorithms
• Product Algorithm:
pre-conditions: m  0, i = 0, product = 0
while (i < m) {
product += x;
i++;
}
post-condition: product = m * x
Correctness of Some Algorithms
• Division Algorithm
pre-conditions: a  0, d > 0, r = a, q = 0
while (r  d) {
r -= d;
q++;
}
post-conditions: a = q * d + r, 0  r < d
Correctness of Some Algorithms
• Euclidean Algorithm
pre-conditions: a > b  0, r = b
while (b > 0) {
r = a mod b;
a = b;
b = r;
}
post-condition: a = gcd(a, b)
Basics of Set Theory
• Set and element are undefined notions in the set
theory and are taken for granted
• Set notation: {1, 2, 3}, {{1, 2}, {3}, {1, 2, 3}}, {1, 2,
3, …}, , {x  R | -3 < x < 6}
• Set A is called a subset of set B, written as A  B,
when x, x  A  x  B. What is negation?
• A is a proper subset of B, when A is a subset of B
and x  B and x  A
• Visual representation of the sets
• Distinction between  and 
Set Operations
• Set a equals set B, iff every element of set A is in set B
and vice versa. (A = B  A  B /\ B  A)
• Proof technique for showing sets equality (example)
• Union of two sets is a set of all elements that belong to
at least one of the sets (notation on board)
• Intersection of two sets is a set of all elements that
belong to both sets (notation on board)
• Difference of two sets is a set of elements in one set, but
not the other (notation on board)
• Complement of a set is a difference between universal
set and a given set (notation on board)
• Examples
Empty Set
• S = {x  R, x2 = -1}
• X = {1, 3}, Y = {2, 4}, C = X  Y
(X and Y are disjoint)
• Empty set has no elements 
• Empty set is a subset of any set
• There is exactly one empty set
• Properties of empty set:
– A   = A, A   = 
– A  Ac = , A  Ac = U
– Uc = , c = U
Set Partitioning
• Two sets are called disjoint if they have no
elements in common
• Theorem: A – B and B are disjoint
• A collection of sets A1, A2, …, An is called
mutually disjoint when any pair of sets from this
collection is disjoint
• A collection of non-empty sets {A1, A2, …, An} is
called a partition of a set A when the union of
these sets is A and this collection consists of
mutually disjoint sets
Power Set
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Power set of A is the set of all subsets of A
Example on board
Theorem: if A  B, then P(A)  P(B)
Theorem: If set X has n elements, then
P(X) has 2n elements (proof in Section 5.3
– will show if have time)
Cartesian Products
• Ordered n-tuple is a set of ordered n
elements. Equality of n-tuples
• Cartesian product of n sets is a set of ntuples, where each element in the n-tuple
belongs to the respective set participating
in the product
Set Properties
• Inclusion of Intersection:
A  B  A and A  B  B
• Inclusion in Union:
A  A  B and B  A  B
• Transitivity of Inclusion:
(A  B  B  C)  A  C
• Set Definitions:
xXYxXyY
xXYxXyY
xX–YxXyY
x  Xc  x  X
(x, y)  X  Y  x  X  y  Y
Set Identities
• Commutative Laws: A  B = A  B and A  B = B  A
• Associative Laws: (A  B)  C = A  (B  C) and (A  B)  C = A  (B  C)
• Distributive Laws:
A  (B  C) = (A  B)  (A  C) and A  (B  C) = (A  B)  (A  C)
• Intersection and Union with universal set: A  U = A and A  U = U
• Double Complement Law: (Ac)c = A
• Idempotent Laws: A  A = A and A  A = A
• De Morgan’s Laws: (A  B)c = Ac  Bc and (A  B)c = Ac  Bc
• Absorption Laws: A  (A  B) = A and A  (A  B) = A
• Alternate Representation for Difference: A – B = A  Bc
• Intersection and Union with a subset: if A  B, then A  B = A and A  B = B
Proving Equality
• First show that one set is a subset of
another (what we did with examples
before)
• To show this, choose an arbitrary
particular element as with direct proofs
(call it x), and show that if x is in A then x is
in B to show that A is a subset of B
• Example (step through all cases)
Disproofs, Counterexamples and
Algebraic Proofs
• Is is true that (A – B)  (B – C) = A – C?
(No via counterexample)
• Show that (A  B) – C = (A – C)  (B – C)
(Can do with an algebraic proof, slightly
different)
Boolean Algebra
• A Boolean Algebra is a set of elements
together with two operations denoted as +
and * and satisfying the following
properties:
Commutative: a + b = b + a, a * b = b * a
Associative: (a + b) + c = a + (b + c), (a * b) *c = a * (b * c)
Distributive: a + (b * c) = (a + b) * (a + c), a * (b + c) = (a *
b) + (a * c)
Identity: a + 0 = a, a * 1 = a for some distinct unique 0 and
1
Complement: a + ã = 1, a * ã = 0
Russell’s Paradox
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Set of all integers, set of all abstract ideas
Consider S = {A, A is a set and A  A}
Is S an element of S?
Barber puzzle: a male barber shaves all those men who
do not shave themselves. Does the barber shave
himself?
• Consider S = {A  U, A  A}. Is S  S?
• Godel: No way to rigorously prove that mathematics is
free of contradictions. (“This statement is not provable” is
true but not provable) (consistency of an axiomatic
system is not provable within that system)
Halting Problem
• There is no computer algorithm that will
accept any algorithm X and data set D as
input and then will output “halts” or “loops
forever” to indicate whether X terminates
in a finite number of steps when X is run
with data set D.
• Proof is by contradiction