Functions, High-School Edition

Transcription

1 Functions

2 What is a function?

3 Functions, High-School Edition

4 f(x) = x4 5x2 + 4 source:

5 source:

6 Functions, High-School Edition In high school, functions are usually given as objects of the form x3 +3 x2 +15 x+7 f (x) = 1 x137 What does a function do? Takes in as input a real number. Outputs a real number. except when there are vertical asymptotes or other discontinuities, in which case the function doesn't output anything.

7 Functions, CS Edition

8 int int flipuntil(int flipuntil(int n) n) {{ int int numheads numheads == 0; 0; int int numtries numtries == 0; 0; while while (numheads (numheads << n) n) {{ if if (randomboolean()) (randomboolean()) numheads++; numheads++; }} }} numtries++; numtries++; return return numtries; numtries;

9 Functions, CS Edition In programming, functions might take in inputs, might return values, might have side effects, might never return anything, might crash, and might return different values when called multiple times.

10 What's Common? Although high-school math functions and CS functions are pretty different, they have two key aspects in common: They take in inputs. They produce outputs. In math, we like to keep things easy, so that's pretty much how we're going to define a function.

11 Rough Idea of a Function: A function is an object f that takes in one input and produces exactly one output. x f f(x) (This is not a complete definition we'll revisit this in a bit.)

12 High School versus CS Functions In high school, functions usually were given by a rule: f(x) = 4x + 15 In CS, functions are usually given by code: int factorial(int n) { int result = 1; for (int i = 1; i <= n; i++) { result *= i; } return result; } What sorts of functions are we going to allow from a mathematical perspective?

13 Dikdik Nubian Ibex Sloth

14

15 but also

16 f(x) = x2 + 3x 15

17 { n/2 if n is even f (n)= (n+1)/2 otherwise Functions Functionslike likethese these are arecalled calledpiecewise piecewise functions. functions.

18 To define a function, you will typically either draw a picture, or give a rule for determining the output.

19 In mathematics, functions are deterministic. That is, given the same input, a function must always produce the same output.

20 One Challenge

21 f(x) = x2 + 2x + 5 f( 3 ) = = 20 f( 0 ) = = 5 f( 3 ) =?

22 f( )= f(137 ) =?

23 We need to make sure we can't apply functions to meaningless inputs.

24 Domains and Codomains Every function f has two sets associated with it: its domain and its codomain. A function f can only be applied to elements of its domain. For any x in the domain, f(x) belongs to the codomain. Domain Codomain

25 Domains and Codomains If f is a function whose domain is A and whose codomain is B, we write f : A B. This notation just says what the domain and codomain of the function is. It doesn't say how the function is evaluated. Think of it like a function prototype in C or C++. The notation f : A B is like writing B f(a argument); We know that f takes in an A and returns a B, but we don't know exactly which B it's going to return for a given A.

26 Domains and Codomains A function f must be defined for every element of the domain. For example, if f : ℝ ℝ, then the following function is not a valid choice for f: f(x) = 1 / x The output of f on any element of its domain must be an element of the codomain. For example, if f : ℝ ℕ, then the following function is not a valid choice for f: f(x) = x However, a function f does not have to produce all possible values in its codomain. For example, if f : ℕ ℕ, then the following function is a valid choice for f: f(n) = n2

27 Defining Functions Typically, we specify a function by describing a rule that maps every element of the domain to some element of the codomain. Examples: f(n) = n + 1, where f : ℤ ℤ f(x) = sin x, where f : ℝ ℝ f(x) = x, where f : ℝ ℤ Notice that we're giving both a rule and the domain/codomain.

28 Defining Functions Typically, we specify a function by describing a rule that maps every element This function of the domain to some element of the This isis the the ceiling ceiling function the the smallest smallest integer integer greater greater codomain. Examples: than than or or equal equal to to x. x. For For example, example, 1 1 == 1,1, == 2, 2, f(n) = n + 1, where f : ℤ ℤ and and π π == f(x) = sin x, where f : ℝ ℝ f(x) = x, where f : ℝ ℤ Notice that we're giving both a rule and the domain/codomain.

29 Is this a function from A to B? Stanford Cardinal Berkeley Blue Michigan Gold Arkansas White A B

30 Is this a function from A to B? California Dover New York Sacramento Delaware Albany Washington DC A B

31 Is this a function from A to B? Wish Funshine A B Love-a-Lot Tenderheart Friend

32 Combining Functions

33 f : People Places Keith g : Places Prices Mountain View Far Too Much Erik San Francisco King's Ransom Kevin Redding, CA A Modest Amount Cagla Barrow, AK Pocket Change Andi People Palo Alto Places Prices

34 f : People Places Keith g : Places Prices Mountain View Far Too Much Erik San Francisco King's Ransom Kevin Redding, CA A Modest Amount Cagla Barrow, AK Pocket Change Andi People Palo Alto Places h : People Prices h(x) = g(f(x)) Prices

35 Keith Far Too Much Erik King's Ransom Kevin A Modest Amount Cagla Pocket Change Andi People Places h : People Prices h(x) = g(f(x)) Prices

36 Function Composition Suppose that we have two functions f : A B and g : B C. Notice that the codomain of f is the domain of g. This means that we can use outputs from f as inputs to g. x f f(x) g g(f(x))

37 Function Composition Suppose that we have two functions f : A B and g : B C. The composition of f and g, denoted g f, is a function where g f : A C, and (g f)(x) = g(f(x)). A few things to notice: The Thename nameofofthe thefunction functionisisgg f.f. When Whenwe weapply applyititto toan aninput inputx,x, we wewrite write(g (g f)(x). f)(x).i Idon't don'tknow know why, but that's what we do. why, but that's what we do. The domain of g f is the domain of f. Its codomain is the codomain of g. Even though the composition is written g f, when evaluating (g f)(x), the function f is evaluated first.

38 Function Composition Let f : ℕ ℕ be defined as f(n) = 2n + 1 and g : ℕ ℕ be defined as g(n) = n2. What is g f? (g f)(n) = g(f(n)) (g f)(n) = g(2n + 1) (g f)(n) = (2n + 1)2 = 4n2 + 4n + 1 What is f g? (f g)(n) = f(g(n)) (f g)(n) = f(n2) (f g)(n) = 2n2 + 1 In general, if they exist, the functions g f and f g are usually not the same function. Order matters in function composition!

39 Time-Out for Announcements!

40 Are Are you you on on aa mailing mailing list list where where this this career career fair fair was was advertised? advertised? If If so, so, let let me me know know which which one! one!

41 WiCS Casual CS Dinner Stanford WiCS (Women in Computer Science) is holding a Casual CS Dinner tonight at 6:00PM at the Women's Community Center. All are welcome. Highly recommended!

42 ostem Mixer Stanford's chapter of ostem (Out in STEM) is holding a mixer event tonight at 6:00PM at the LGBT-CRC. If I'm not mistaken, that's literally right above the Casual CS Dinner! Interested in attending? RSVP using this link.

43 Problem Set One Problem Set One has been graded. Feedback is now available online through GradeScope, and solutions are now available in hardcopy. Please read Handout #15 ( Reviewing Graded Work ) for our advice about what to do next. In particular: Make sure you understand every piece of feedback you received. Ask questions about feedback you don't fully understand. Read over our solution sets, especially the why we asked this question section, to make sure that you understand what skills we were trying to help you build. Late PS1 submissions will be returned by tomorrow afternoon at 3:00PM.

44 Problem Set Two The checkpoint assignments for PS2 have been graded. Please be sure to read over the checkpoint solutions set there's a lot of information in there! The remaining problems are due on Friday. Please stop by office hours with questions, and continue to ask questions on Piazza!

45 Problem Set One: A Common Mistake

46 An Incorrect Proof Theorem: For any integers x, y, and k, if x ₖ y, then y ₖ x. Proof: Consider any arbitrary integers x, y, and k where x ₖ y. This means that there is an integer q where x y = kq. We need to prove that y ₖ x, meaning that we need to prove that there is an integer r where y x = kr. Since y x = kr, we see that x y = -kr. Earlier we noted that x y = kq, so collectively we see that -kr = kq. Therefore, we see that r = -q.

47 An Incorrect Proof Theorem: For any integers x, y, and k, if x ₖ y, then y ₖ x. Proof: Consider any arbitrary integers x, y, and k where x ₖ y. This means that there is an integer q where x y = kq. We need to prove that y ₖ x, meaning that we need to prove that there is an integer r where y x = kr. Since y x = kr, we see that x y = -kr. Earlier we noted that x y = kq, so collectively we see that -kr = kq. Therefore, we see that r = -q. We're We're assuming assuming what what we're we're trying trying to to prove! prove!

48 A Better Proof Theorem: For any integers x, y, and k, if x ₖ y, then y ₖ x. Proof: Consider any arbitrary integers x, y, and k where x ₖ y. This means that there is an integer q where x y = kq. We need to prove that y ₖ x, meaning that we need to prove that there is an integer r where y x = kr. Since x y = kq, we see that y x = -kq = k(-q). Therefore, there is an integer r, namely -q, such that y x = kr. Consequently, we see that y ₖ x, as required. Notice Notice that that we we start start with with our our initial initial assumptions assumptions and and use use them them to to derive derive the the required required result. result.

49 General Advice Be careful not to assume what you're trying to prove. In a proof, we recommend using the phrases we need to show that or we need to prove that to clearly indicate your goals. If you later find yourself relying on a statement marked we need to prove that, chances are you've made an error in your proof.

50 Your Questions

51 What was the best piece of advice you ever received? Both math-related and not. For For math math advice: advice: All All models models are are wrong. wrong. Some Some are are useful. useful. (source) (source) General General life life advice: advice: if if you you can t can t make make your your opponent s opponent s point point for for them, them, you you don t don t truly truly grasp grasp the the issue. issue. (source) (source)

52 You asked us to ask you again about research versus industry over the summer. :-) We're We're lucky lucky to to be be inin aa spot spot where where undergrads undergrads can can get get internships internships and and where where recruiting recruiting isis strong. strong. However, However, II think think it's it's important important that that everyone everyone get get some some research research experience experience as as well. well. Keep Keep inin mind mind that that post-graduation, post-graduation, unless unless you you directly directly go go for for aa Ph.D, Ph.D, it's it's really, really, really really hard hard to to get get into into research. research. It's It's worth worth exploring exploring itit to to see see ifif it's it's something something you're you're interested interested inin before before you you graduate, graduate, even even ifif you you decide decide not not to to continue continue on on inin academia. academia. If If you're you're an an undergraduate undergraduate inin CS, CS, II strongly strongly recommend recommend looking looking into into the the CURIS CURIS summer summer research research program. program. It's It's aa great great way way to to get get exposed exposed to to research, research, make make connections connections inin the the CS CS department, department, and and get get aa feel feel for for what what academia academia isis like. like.

53 Back to CS103!

54 Special Types of Functions

55 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

56 Injective Functions A function f : A B is called injective (or one-to-one) if the following statement is true about f: a₁ A. a₂ A. (a₁ a₂ f(a₁) f(a₂)) ( If the inputs are different, the outputs are different. ) The following definition is equivalent and tends to be more useful in proofs: a₁ A. a₂ A. (f(a₁) = f(a₂) a₁ = a₂) ( If the outputs are the same, the inputs are the same. ) A function with this property is called an injection. Intuitively, in an injection, every element of the codomain has at most one element of the domain mapping to it.

57 Injective Functions Theorem: Let f : ℕ ℕ be defined as f(n) = 2n + 7. Then f is injective. Proof: Consider any n₀, n₁ ℕ where f(n₀) = f(n₁). We will prove that n₀ = n₁. What What does does itit mean mean for for the the function function ff to to be be Since f(n₀) = f(n₁), we see that injective? injective? 2n₀ + 7 = 2n₁ + 7. This in turn means that n₀ ℕ. n₁ ℕ. n₀ ℕ. n₁ ℕ.((f(n₀) f(n₀)==f(n₁) f(n₁) n₀ n₀==n₁ n₁)) 2n₀ = 2n₁, n₀ n₀ ℕ. ℕ. n₁ n₁ ℕ. ℕ.((n₀ n₀ n₁ n₁ f(n₀) f(n₀) f(n₁) f(n₁))) so n₀ = n₁, as required. Therefore, Therefore, we'll we'll pick pick arbitrary arbitrary n₀, n₀,n₁ n₁ ℕ ℕ where where f(n₀) f(n₀)==f(n₁), f(n₁), then then prove prove that that n₀ n₀==n₁. n₁.

58 Injective Functions Theorem: Let f : ℕ ℕ be defined as f(n) = 2n + 7. Then f is injective. Proof: Consider any n₀, n₁ ℕ where f(n₀) = f(n₁). We will prove that n₀ = n₁. Since f(n₀) = f(n₁), we see that 2n₀ + 7 = 2n₁ + 7. This in turn means that 2n₀ = 2n₁, so n₀ = n₁, as required.

59 Injective Functions Theorem: Let f : ℤ ℕ be defined as f(x) = x4. Then f is not injective. Proof: We will prove that there exist integers x₀ and x₁ such that x₀ x₁, but f(x₀) = f(x₁). What does it mean for f to be injective? What does it mean for f to be injective? Let x₀ = -1 and x₁ = +1. Then x₀ ℤ. x₁ ℤ. (x₀ x₀ ℤ. x₁ ℤ. (x₀ x₁ x₁ f(x₀) f(x₀) f(x₁)) f(x₁)) 4 f(x₀) f(-1) = (-1) =1 What is the negation of this = statement? What is the negation of this statement? and x₀ x₀ ℤ. ℤ. x₁ x₁ ℤ. ℤ.(x₀ (x₀ x₁ x₁ f(x₀) f(x₀) f(x₁)) f(x₁)) x₀ ℤ. = x₁ 4 f(x₀) x₀ ℤ. ℤ. x₁ x₁ ℤ. (x₀ x₁1 f(x₁)) f(x₁) =(x₀ f(1) =f(x₀) 1, f(x₁)) x₀ x₀ ℤ. ℤ. x₁ x₁ ℤ. ℤ. (x₀ (x₀ x₁ x₁ f(x₀) f(x₀) f(x₁)) f(x₁)) so f(x₀) = f(x₁) even though x₀ x₁, as required. x₀ x₀ ℤ. ℤ. x₁ x₁ ℤ. ℤ.(x₀ (x₀ x₁ x₁ (f(x₀) (f(x₀) f(x₁))) f(x₁))) x₀ x₀ ℤ. ℤ. x₁ x₁ ℤ. ℤ.(x₀ (x₀ x₁ x₁ f(x₀) f(x₀)==f(x₁)) f(x₁)) Therefore, we need to find x₀, x₁ ℤ such that x₀ x₁, but f(x₀) = f(x₁). Therefore, we need to find x₀, x₁ ℤ such that x₀ x₁, but f(x₀) = f(x₁). Can we do that? Can we do that?

60 Injective Functions Theorem: Let f : ℤ ℕ be defined as f(x) = x4. Then f is not injective. Proof: We will prove that there exist integers x₀ and x₁ such that x₀ x₁, but f(x₀) = f(x₁). Let x₀ = -1 and x₁ = +1. Then f(x₀) = f(-1) = (-1)4 = 1 and f(x₁) = f(1) = 14 = 1, so f(x₀) = f(x₁) even though x₀ x₁, as required.

61 Injections and Composition

62 Injections and Composition Theorem: If f : A B is an injection and g : B C is an injection, then the function g f : A C is an injection. Our goal will be to prove this result. To do so, we're going to have to call back to the formal definitions of injectivity and function composition.

63 Theorem: If f : A B is an injection and g : B C is an injection, then the function g f : A C is also an injection. Proof: Let f : A B and g : B C be arbitrary injections. We will prove that the function g f : A C is also injective. To do so, we will prove for all a₁ A and a₂ A that if (g f)(a₁) = (g f)(a₂), then a₁ = a₂. Suppose that (g f)(a₁) = (g f)(a₂). Expanding out the definition of g f, this means that g(f(a₁)) = g(f(a₂)). Since g is injective and g(f(a₁)) = g(f(a₂)), we know f(a₁) = f(a₂). Similarly, since f is injective and f(a₁) = f(a₂), we know that a₁ = a₂, as required. Good Good exercise: exercise: Repeat Repeat this this proof proof using using the the other other definition definition of of injectivity. injectivity.

64 Another Class of Functions

65 Front Door Balcony Window Bedroom Window

66 Surjective Functions A function f : A B is called surjective (or onto) if this statement is true about f: b B. a A. f(a) = b ( For every possible output, there's at least one possible input that produces it ) A function with this property is called a surjection. Intuitively, every element in the codomain of a surjection has at least one element of the domain mapping to it.

67 Surjective Functions Theorem: Let f : ℝ ℝ be defined as f(x) = x / 2. Then f(x) is surjective. Proof: Consider any y ℝ. We will prove that there is a choice of x ℝ such that f(x) = y. Let x = 2y. Then we see that f(x) = f(2y) = 2y / 2 = y. So f(x) = y, as required.

68 Composing Surjections

69 Theorem: If f : A B is surjective and g : B C is surjective, then g f : A C is also surjective. Proof: Let f : A B and g : B C be arbitrary surjections. We will prove that the function g f : A C is also surjective. To do so, we will prove that for any c C, there is some a A such that (g f)(a) = c. Equivalently, we will prove that for any c C, there is some a A such that g(f(a)) = c. b Consider any c C. Since g : B C is surjective, there is a b B such that g(b) = c. Similarly, since f : A B is some c surjective, there is some a A such that f(a) = b. This means that there is some a A such that g(f(a)) = g(b) = c, which is what we needed to show. A f B g C

70 Theorem: If f : A B is surjective and g : B C is surjective, then g f : A C is also surjective. Proof: Let f : A B and g : B C be arbitrary surjections. We will prove that the function g f : A C is also surjective. To do so, we will prove that for any c C, there is some a A such that (g f)(a) = c. Equivalently, we will prove that for any c C, there is some a A such that g(f(a)) = c. Consider any c C. Since g : B C is surjective, there is some b B such that g(b) = c. Similarly, since f : A B is surjective, there is some a A such that f(a) = b. This means that there is some a A such that g(f(a)) = g(b) = c, which is what we needed to show.