Transcript Lecture_5 - New York University

Discrete Mathematics
Lecture 5
Harper Langston
New York University
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
Counting and Probability
• Coin tossing
• Random process
• Sample space is the set of all possible outcomes
of a random process. An event is a subset of a
sample space
• Probability of an event is the ratio between the
number of outcomes that satisfy the event to the
total number of possible outcomes
P(E) = N(E)/N(S) for event E and sample space
S
• Rolling a pair of dice and card deck as sample
random processes
Possibility Trees
• Teams A and B are to play each other
repeatedly until one wins two games in a
row or a total three games.
– What is the probability that five games will be
needed to determine the winner?
• Suppose there are 4 I/O units and 3
CPUs. In how many ways can I/Os and
CPUs be attached to each other when
there are no restrictions?
Multiplication Rule
• Multiplication rule: if an operation consists of k
steps each of which can be performed in ni ways
(i = 1, 2, …, k), then the entire operation can be
performed in ni ways.
• Number of PINs
• Number of elements in a Cartesian product
• Number of PINs without repetition
• Number of Input/Output tables for a circuit with n
input signals
• Number of iterations in nested loops
Multiplication Rule
• Three officers – a president, a treasurer
and a secretary are to be chosen from four
people: Alice, Bob, Cindy and Dan. Alice
cannot be a president, Either Cindy or Dan
must be a secretary. How many ways can
the officers be chosen?
Permutations
• A permutation of a set of objects is an ordering
of these objects
• The number of permutations of a set of n objects
is n! (Examples)
• An r-permutation of a set of n elements is an
ordered selection of r elements taken from a set
of n elements: P(n, r) (Examples)
• P(n, r) = n! / (n – r)!
• Show that P(n, 2) + P(n, 1) = n2
Addition Rule
• If a finite set A is a union of k mutually disjoint
sets A1, A2, …, Ak, then n(A) = n(Ai)
• Number of words of length no more than 3
• Number of 3-digit integers divisible by 5
Difference Rule
• If A is a finite set and B is its subset, then n(A –
B) = n(A) – n(B)
• How many PINS contain repeated symbols?
• So, P(Ac) = 1 – P(A) (Example for PINS)
• How many students are needed so that the
probability of two of them having the same
birthday equals 0.5?
Inclusion/Exclusion Rule
• Page 327 for 2 sets
• 3 sets
Combinations
• An r-combination of a set of n elements is a
subset of r elements: C(n, r)
• Permutation is an ordered selection,
combination is an unordered selection
• Quantitative relationship between permutations
and combinations: P(n, r) = C(n, r) * r!
• Permutations of a set with repeated elements
• Double counting
Team Selection Problems
• There are 12 people, 5 men and 7 women, to
work on a project:
– How many 5-person teams can be chosen?
– If two people insist on working together (or not working
at all), how many 5-person teams can be chosen?
– If two people insist on not working together, how many
5-person teams can be chosen?
– How many 5-person teams consist of 3 men and 2
women?
– How many 5-person teams contain at least 1 man?
– How many 5-person teams contain at most 1 man?
Poker Problems
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What is a probability to contain one pair?
What is a probability to contain two pairs?
What is a probability to contain a triple?
What is a probability to contain royal flush?
What is a probability to contain straight flush?
What is a probability to contain straight?
What is a probability to contain flush?
What is a probability to contain full house?
Combinations with Repetition
• An r-combination with repetition allowed is
an unordered selection of elements where
some elements can be repeated
• The number of r-combinations with
repetition allowed from a set of n elements
is C(r + n –1, r)
• Soft drink example
Algebra of Combinations and
Pascal’s Triangle
• The number of r-combinations from a set
of n elements equals the number of (n – r)combinations from the same set.
• Pascal’s triangle: C(n + 1, r) = C(n, r – 1) +
C(n, r)
• C(n,r) = C(n,n-r)
Binomial Formula
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(a + b)n = C(n, k)akbn-k
Show that C(n, k) = 2n
Show that (-1)kC(n, k) = 0
Express kC(n, k)3k in the closed form