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Week 7 - Wednesday

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What did we talk about last time? Set proofs and disproofs Russell’s paradox

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Imagine you are in a completely dark room with a deck of cards containing 10 cards that are face up and the rest face down How can you create two piles of cards (not necessarily the same height) that both contain the same number of face up cards? If you fail, you'll be eaten by a grue

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Prove that, for any set A, A = Hint: Use a proof by contradiction

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For all sets A, B, and C, if A B and B C, then A C

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For all sets A and B, ((A c B c ) – A) c = A

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Set theory is a slippery slope We are able to talk about very abstract concepts { x Z | x is prime } This is a well-defined set, even though there are an infinite number of primes and we don't know how to find the n th prime number Without some careful rules, we can begin to define sets that are not well-defined

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Let a barber be the man in Elizabethtown who shaves the men in Elizabethtown if and only if they don't shave themselves. Let T be the set of all men in Elizabethtown Let B(x) be "x is a barber" Let S(x,y) be "x shaves y" b T m T (B(b) (S(b,m) ~S(m,m))) But, who shaves the barber?

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Bertrand Russell invented the Barber Paradox to explain to normal people a problem he had found in set theory Most sets are not elements of themselves So, it seems reasonable to create a set S that is the set of all sets that are not elements of themselves More formally, S = { A | A is a set and A A } But, is S an element of itself?

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How do we make sure that this paradox cannot happen in set theory? We can make rules about what sets we allow in or not The rule that we use in class is that all sets must be subsets of a defined universe U Higher level set theory has a number of different frameworks for defining a useful universe

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It turns out that the idea behind Russell's Paradox actually has practical implications It wasn't new, either Cantor had previously used a diagonal argument to show that there are more real numbers than rational numbers But, unexpectedly, Turing found an application of this idea for computing

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A Turing machine is a mathematical model for computation It consists of a head, an infinitely long tape, a set of possible states, and an alphabet of characters that can be written on the tape A list of rules saying what it should write and should it move left or right given the current symbol and state 1011110000 A A

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If an algorithm exists, a Turing machine can perform that algorithm In essence, a Turing machine is the most powerful model we have of computation Power, in this sense, means the ability to compute some function, not the speed associated with its computation

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Given a Turing machine and input x, does it reach the halt state? First, recognize that we can encode a Turing machine as input for another Turing machine We just have to design a system to describe the rules, the states, etc. We want to design a Turing machine that can read another

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Imagine we have a Turing machine H(m,x) that takes the description of another Turing machine m and its input x and returns 1 if m halts on input x and 0 otherwise Now, construct a machine H’(m,x) that runs H(m,x), but, if H(m,x) gives 1, then H’(m,x) infinitely loops, and, if H(m,x) gives 0, then then H’(m,x) returns 1 Let’s say that d is the description of H’(m,x) What happens when you run H’(d,d)?

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Clearly, a Turing machine that solves the halting problem doesn’t exist Essentially, the problem of deciding if a problem is computable is itself uncomputable Therefore, there are some problems (called undecidable) for which there is no algorithm Not an algorithm that will take a long time, but no algorithm If we find such a problem, we are stuck …unless someone can invent a more powerful model of computation

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Gödel used diagonalization again to prove that it is impossible to create a consistent set of axioms that can prove everything about the set of natural numbers As a consequence, you can create a system that is complete but not consistent Or, you can create a system that is consistent but not complete Either way, there are principles in math in general that are true but impossible to prove, at least with any given system You might as well give up on math now

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Student Lecture

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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 }

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Using standard assumptions, consider f(x) = x 2 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?

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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 abcabc abcabc 12341234 12341234 XYf

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Which of the following are functions from X to Y? abcabc abcabc 12341234 12341234 XYf abcabc abcabc 12341234 12341234 XYg abcabc abcabc 12341234 12341234 XYh

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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?

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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

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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 + to R as follows: log b x = y b y = x

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For a function of multiple values, we can define its domain to be the Cartesian product of sets Let S n 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: S n x S n Z nonneg What is H(00101, 01110)? What is H(10001, 01111)?

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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

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Let F be a function from X to Y F is one-to-one (or injective) if and only if: If F(x 1 ) = F(x 2 ) then x 1 = x 2 Is f(x) = x 2 from Z to Z one-to-one? Is f(x) = x 2 from Z + to Z one-to-one? Is h(x) one-to-one? abcabc abcabc 12341234 12341234 XYh

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To prove that f from X to Y is one-to-one, prove that x 1, x 2 X, f(x 1 ) = f(x 2 ) x 1 = x 2 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) = n 2 is not one-to-one

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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) = x 2 from Z to Z onto? Is f(x) = x 2 from R + to R + onto? Is h(x) onto? abcabc abcabc 123123 123123 XYh

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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

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More on functions One to one and onto Cardinality

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Work on Homework 5 Due the Monday after Spring Break Keep reading Chapter 7

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