Understanding Functions: Definitions, Properties, and Practical Applications

1. Week 7 - Friday CS322

2. Last time • What did we talk about last time? • Set disproofs • Russell’s paradox • Function basics

3. Questions?

4. 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?

5. Functions

6. 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 }

7. 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?

8. 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

9. 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

10. 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?

11. 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

12. 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

13. 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)?

14. 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

15. 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

16. 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

17. 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

18. 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

19. Composition of Functions

20. 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

21. 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

22. 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)?

23. 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?

24. 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?

25. Pigeonhole Principle Student Lecture

26. Pigeonhole Principle

27. 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

28. 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?

29. 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

30. Cardinality

31. 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

32. 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

33. 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

34. 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

35. 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)

36. Upcoming

37. Next time… • Relations (after Spring Break) • Exam 2 is the Monday after the Monday after Spring Break

38. Reminders • Work on Homework 5 • Due on Monday after Spring Break • Look at Homework 6 • Read Chapter 8 for after Spring Break