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## CS322

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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

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 xX
- 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

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