Week 7 - Friday. What did we talk about last time? Set disproofs Russell’s paradox Function basics.
Post on 01-Jan-2016
220 Views
Preview:
Transcript
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
a
b
c
1
2
3
4
X Yf
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?
2x
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: logb x = 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 YF 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?
a
b
c
1234
X Yh
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 YF 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?
a
b
c
123
X Yh
Inverse functions
If a function F: X Y is 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)?
1
2
3
e
x
y
z
a
b
c
d
f g
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
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 B iff 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
1/1 1/2 1/3 1/4
2/1 2/2 2/3 2/4
3/1 3/2 3/3 3/4
4/1 4/2 4/3 4/4
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)
top related