Download “No professor has been asked questions by all of his students

Section 2.5 Contradiction and the Pigeonhole Principle
In a direct proof we assume
and prove
In a contrapositive proof we assume
In a proof by contradiction we assume
and prove
and
Read propositions 1 – 3.
Theorem 4: The real number 2 is irrational. That is, there does
not exist a rational number r such that r2 = 2.
Definition: Two numbers are relatively prime if they have no
common divisor greater than 1.
Proof that 2 is irrational
Constructive and Nonconstructive Proofs
Proving a statement of the type x P(x)
constructive: Find a specific value of x that makes the statement true.
Example: For every integer n there is an even integer x such that x > n
In a nonconstructive proof
Example: For every positive integer n there is a prime number p
such that p > n
Difference between contradiction and contrapositive proofs
Prove that if n is an integer and n3 + 5 is odd, then n is even.
Contrapositive Proof: Suppose n is odd.
Proof by contradiction:
Another example: If a number added to itself gives itself, then the
number is 0.
Frequently, a direct or contrapositive proof is hiding in a
supposed proof by contradiction.
The Pigeonhole Principle
Pigeonhole principle:
Example 1: In any set of five numbers, there are two numbers x
and y that have the same remainder when divided by 4.
Example 2: If five numbers are chosen from the set {1, 2, 3, 4, 5,
6, 7, 8}, then two of the numbers chosen must have a sum of 9.
The generalized pigeonhole principle:
Example 3:
Example 4:
Example 5:
More Examples
Example 6: Given a group of six acquaintances prove there are 3
mutual friends or 3 mutual enemies.
Example 7: How many students each of whom comes from one of
the 50 states must be enrolled at a university to guarantee there
are at least 100 who come from the same state?
Example 8: How many people must be in a room to guarantee
that two people have the same first and last initials?
Section 2.6 Excursion: Representations of Numbers
Properties of numbers that are independent of the base:
Definition: Given a positive integer X, the decimal representation for
X is a string consisting of the digits from {0, 1, 2, …, 9} that looks
like dndn-1…d2d1d0 where dn•10n + dn-1•10n-1+ … + d1•101 + d0•100 = X.
Definition: The base two (binary representation) of a positive
integer X is a string consisting of digits from {0, 1} that looks like
bnbn-1…b2b1b0 where bn•2n + bn-1•2n-1+ … + b1•21 + b0•20 = X.
Proposition 4: A natural number is divisible by 9 iff the sum of its
(decimal) digits is divisible by 9.
Converting from one base to another
Convert 236 from base 10 to base 2
The numerals used in base k are {0, 1, …, k-1} if k  10.
For bases higher than ten,
Algorithm for converting a natural number to base k
input a natural number n
while n > 0 do
divide n by k and get a quotient q and remainder r
write r as the next digit in right to left order
replace the value of n with q and repeat
Converting a base k number to base 10
Example: Convert (3B6)hex to base 10
2.7 Excursion: Modular Arithmetic and Cryptography
Defn. If a and b are integers and n is a positive integer > 1 then a
is congruent to b modulo n iff
Notation: a  b (mod n) or a n b
Theorem 1: For integers a, b and n with n  2, a mod n = b mod n
iff n | (a-b). That is, a n b iff a mod n = b mod n.
Theorem 2: Consider an integer n  2. For all integers a, b, c and d,
if a n b and c n d then
1. (a•c) n (b•d)
2. (a + c) n (b + d)
GCD and related concepts
The greatest common divisor of two integers a and b
Example: gcd(18, 24)
If gcd(a, b) = 1, then a and b are relatively prime.
The least common multiple of integers a and b is
Example: lcm(18, 24)
Using prime factorization to find the gcd and the lcm:
The Euclidean Algorithm
Finding the greatest common divisor of a and b
Notation: gcd(a, b)
find gcd (662, 414)
662 =
find gcd(330, 156)
330 =
function gcd(a, b: positive integers)
xa
yb
while y  0
r  x mod y
xy
yr
return(x)
Chapter 3 Sets and Boolean Algebra
Special Sets (defined earlier): Z, N, Z+,Q,R
Methods of describing sets:
Notational conventions:
1. The notation x  A
2. A is a subset of B (written A  B)
2 ½. A is a proper subset of B written A  B
3. A = B if A  B and B  A
Proving sets equal
4. The empty set denoted { } or  is the set with no elements
Example: Let S be any set. Which of the following are true?
SS
SS
ØS
ØS
5. Universal set
Sets containing other sets. Let C = {a, b, Ø, {e, f} }.
Which of the following statements is or are true?
{e, f}  C
{e, f}  C
The power set of a set A is
Notation:
The cardinality or size of a set A
Notation:
A set is finite iff
Example: Find smallest set S such that {1, 2} is both an element
of S and a subset of S.
Set Operations
Complement of A
Alternate notations:
union:
Intersection:
Two sets A and B are disjoint if
set difference:
symmetric difference:
Set Identities
Union
Intersection
AB=BA
AB=BA
Name
Commutative
(A  B)  C = A  (B  C) (AB)  C = A  (B  C) Associative
A  (B  C) =
(A  B)  (A  C)
A   = A, A  U = U
A  A' = U
A  (B  C) =
(A  B)  (A  C)
A   = , A  U = A
A  A' = 
(A  B)' = A'  B'
(A  B)' = A'  B'
Distributive
Idempotent,
Domination
Laws
Complement
DeMorgan's
Laws
A'  B' if and only if B  A
A  B if and only if A  B = B
A  B if and only if A  B = A
Identities for Other Set Operations
A'' = A
A B = (A  B) - (A  B)
A A = 
' = U, U' = 
A A' = U
A B = (A - B)  (B - A)
(A B) C = A B C)
A  (A  B) = A
A  (A  B) = A
The Inclusion Exclusion Principle
Theorem 2 (Inclusion-Exclusion Principle) Let A, B, and C be sets
1. n(A  B) =
Let A = {1, 3, 5, 7, 9} and let B = {4, 5, 6, 7, 8}
Two special cases:
2. n(A  B  C) =
Example
The students in a class of size 50 were asked to tell which of three
kinds of pies they liked—apple, cherry or banana cream. The
following results were obtained:
20 liked apple
28 liked cherry
16 liked banana cream
13 liked both apple and cherry
6 liked both apple and banana cream
11 liked both cherry and banana cream
4 liked all three
How many liked none of the three kinds of pies?
How many liked exactly one kind?
How many liked two kinds?
How many liked cherry or banana cream?
How many liked apple but not banana?
Proving Set Identities
(Element-wise proofs)
Example: Proof of one of DeMorgan's Laws: (A  B)' = A'  B'
1st part: Prove (A  B)'  A' B'
Let x  (A  B)'
2nd part: Prove A'  B'  (A  B)'
Let y  A'  B'
Section 3.2 More Operations on Sets
Products and Tuples
Equality of tuples: (a1, a2, …, an) = (b1, b2, …, bn) iff
Cartesian product
Example: Let A = {1, @, #} and let B = {x, y}.
AB=
BA=
Cardinality of a Cartesian Product
Proof Example
Prove that for nonempty sets A and B, if A = B then A  B = B  A
Part 1: Prove A  B  B  A
Part 2: Prove B  A  A  B
Generalizing the Definitions
For n  3, the structure (x1, x2, …, xn) is called an n-tuple
The Cartesian product S1  S2  …  Sn is
Theorem 2: For finite sets S1, S2, …, Sn,
n(S1  S2  …  Sn ) =
A partition of a set S is a collection of disjoint nonempty subsets
of S whose union is S.
Example: Consider mod 5
Section 3.3 Proving Set Properties
Example 1: Prove A  B if and only if A  B = A
Example 2: Prove A - B = A  B′