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Week 7 - Friday. CS322. Last time. What did we talk about last time? Set disproofs Russell’s paradox Function basics. Questions?. Logical warmup. A man has two 10 gallon jars The first contains 6 gallons of wine and the second contains 6 gallons of water

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## CS322

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Week 7 - Friday

### CS322

Last time
• What did we talk about last time?
• Set disproofs
• Russell’s paradox
• Function basics
Logical warmup
• A man has two 10 gallon jars
• The first contains 6 gallons of wine and the second contains 6 gallons of water
• He poured 3 gallons of wine into the water jar and stirred
• Then he poured 3 gallons of the mixture in the water jar into the wine jar and stirred
• Then he poured 3 gallons of the mixture in the wine jar into the water jar and stirred
• He continued the process until both jars had the same concentration of wine
• How many pouring operations did he do?
Definitions
• A function f from set X to set Y is a relation between elements of X (inputs) and elements of Y (outputs) such that each input is related to exactly one output
• We write f: X Y to indicate this
• X is called the domain of f
• Y is called the co-domain of f
• The range of f is { y  Y | y = f(x), for some x  X}
• The inverse image of y is { x  X | f(x) = y }
Examples
• Using standard assumptions, consider f(x) = x2
• What is the domain?
• What is the co-domain?
• What is the range?
• What is f(3.2)?
• What is the inverse image of 4?
• Assume that the set of positive integers is the domain and co-domain
• What is the range?
• What is f(3.2)?
• What is the inverse image of 4?
Arrow diagrams
• With finite domains and co-domains, we can define a function using an arrow diagram
• What is the domain?
• What is the co-domain?
• What are f(a), f(b), and f(c)?
• What is the range?
• What are the inverse images of 1, 2, 3, and 4?
• Represent f as a set of ordered pairs

X

f

Y

1

2

3

4

a

b

c

Functions?
• Which of the following are functions from X to Y?

X

f

Y

1

2

3

4

a

b

c

X

g

Y

X

h

Y

1

2

3

4

1

2

3

4

a

b

c

a

b

c

Function equality
• Given two functions f and g from X to Y,
• f equals g, written f = g, iff:
• f(x) = g(x) for all xX
• Let f(x) = |x| and g(x) =
• Does f = g?
• Let f(x) = x and g(x) = 1/(1/x)
• Does f = g?
Applicability of functions
• Functions can be defined from any well-defined set to any other
• There is an identity function from any set to itself
• We can represent a sequence as a function from a range of integers to the values of the sequence
• We can create a function mapping from sets to integers, for example, giving the cardinality of certain sets
Logarithms
• You should know this already
• But, this is the official place where it should be covered formally
• There is a function called the logarithm with base b of x defined from R+ - {1}to R as follows:
• logbx = y by = x
Functions defined on Cartesian products
• For a function of multiple values, we can define its domain to be the Cartesian product of sets
• Let Sn be strings of 1's and 0's of length n
• An important CS concept is Hamming distance
• Hamming distance takes two binary strings of length n and gives the number of places where they differ
• Let Hamming distance be H: Sn x Sn Znonneg
• What is H(00101, 01110)?
• What is H(10001, 01111)?
Well-defined functions
• There are two ways in which a function can be poorly defined
• It does not provide a mapping for every value in the domain
• Example: f: R R such that f(x) = 1/x
• It provides more than one mapping for some value in the domain
• Example: f: Q Z such that f(m/n) = m, where m and n are the integers representing the rational number
One-to-one functions
• Let F be a function from X to Y
• F is one-to-one (or injective) if and only if:
• If F(x1) = F(x2) then x1 = x2
• Is f(x) = x2 from Z to Z one-to-one?
• Is f(x) = x2 from Z+ to Z one-to-one?
• Is h(x) one-to-one?

X

h

Y

1

2

3

4

a

b

c

Proving one-to-one
• To prove that f from X to Y is one-to-one, prove that  x1, x2  X, f(x1) = f(x2)  x1 = x2
• To disprove, just find a counter example
• Prove that f: R  R defined by f(x) = 4x – 1 is one-to-one
• Prove that g: Z  Z defined by g(n) = n2 is not one-to-one
Onto functions
• Let F be a function from X to Y
• F is onto (or surjective) if and only if:
• y  Y, x  X such that F(x) = y
• Is f(x) = x2 from Z to Z onto?
• Is f(x) = x2 from R+ to R+ onto?
• Is h(x) onto?

X

h

Y

1

2

3

a

b

c

Inverse functions
• If a function F: X Yis both one-to-one and onto (bijective), then there is an inverse function F-1: Y X such that:
• F-1(y) = x F(x) = y, for all x X and y  Y
Composition of functions
• If there are two functions f: A B and g: Y  Z such that the range of f is a subset of the domain of g, we can define a new function g o f: A  Z such that
• (g o f)(x) = g(f(x)), for all x  A
Finite sets
• As before, we can show these functions for finite sets using arrow diagrams
• What's the arrow diagram for (g o f)(x)?

g

f

e

a

b

c

d

1

2

3

x

y

z

Identity function
• The identity function (on set X) maps elements from set X to themselves
• Thus, the identity function ix: X X is:
• iX(x) = x
• For functions f: X  Y and g: Y  X
• What is (f o iX)(x)?
• What is (iX 0 g)(x)?
One-to-one and onto
• If functions f: X  Y and g: Y  Z are both one-to-one, then g o f is one-to-one
• If functions f: X  Y and g: Y  Z are both onto, then g o f is onto
• How would you go about proving these claims?
Inverses
• If f: X Y is one-to-one and onto with inverse function f-1: Y  X, then
• What is f-1 o f?
• What is f o f-1?
Pigeonhole Principle

Student Lecture

Pigeonhole principle
• If n pigeons fly into m pigeonholes, where n > m, then there is at least one pigeonhole with two or more pigeons in it
• More formally, if a function has a larger domain than co-domain, it cannot be one-to-one
• We cannot say exactly how many pigeons are in any given holes
• Some holes may be empty
• But, at least one hole will have at least two pigeons
Pigeonhole examples
• A sock drawer has white socks, black socks, and red argyle socks, all mixed together,
• What is the smallest number of socks you need to pull out to be guaranteed a matching pair?
• Let A = {1, 2, 3, 4, 5, 6, 7, 8}
• If you select five distinct elements from A, must it be the case that some pair of integers from the five you selected will sum to 9?
Generalized pigeonhole principle
• If n pigeons fly into m pigeonholes, and for some positive integer k, n> km, then at least one pigeonhole contains k + 1 or more pigeons in it
• Example:
• In a group of 85 people, at least 4 must have the same last initial
Cardinality
• Cardinality gives the number of things in a set
• Cardinality is:
• Reflexive:A has the same cardinality as A
• Symmetric: If A has the same cardinality as B, B has the same cardinality as A
• Transitive: If A has the same cardinality as B, and B has the same cardinality as C, A has the same cardinality as C
• For finite sets, we could just count the things to determine if two sets have the same cardinality
Cardinality for infinite sets
• Because we can't just count the number of things in infinite sets, we need a more general definition
• For any sets A and B, A has the same cardinality as Biff there is a bijective mapping A to B
• Thus, for any element in A, it corresponds to exactly one element in B, and everything in B has exactly one corresponding element in A
Cardinality example
• Show that the set of positive integers has the same cardinality as the set of all integers
• Hint: Create a bijective function from all integers to positive integers
• Hint 2: Map the positive integers to even integers and the negative integers to odd integers
Countability
• A set is called countably infinite if it has the same cardinality as Z+
• You have just shown that Z is countable
• It turns out that (positive) rational numbers are countable too, because we can construct a table of their values and move diagonally across it, numbering values, skipping numbers that have been listed already
Uncountability
• We showed that positive rational numbers were countable, but a trick similar to the one for integers can show that all rational numbers are countable
• The book gives a classic proof that real numbers are not countable, but we don't have time to go through it
• For future reference, the cardinality of positive integers, countable infinity, is named 0 (pronounced aleph null)
• The cardinality of real numbers, the first uncountable infinity (because there are infinitely many uncountable infinities), is named 1 (pronounced aleph 1)
Next time…
• Relations (after Spring Break)
• Exam 2 is the Monday after the Monday after Spring Break
Reminders
• Work on Homework 5
• Due on Monday after Spring Break
• Look at Homework 6
• Read Chapter 8 for after Spring Break