CPSC 121: Models of Computation PART 1 REVIEW OF TEXT READING Unit 12: Functions These pages correspond to text reading and are not covered in the lectures. Based on slides by Patrice Belleville and Steve Wolfman Unit 12: Functions 2 What is a Function? Plotting Functions f(x) = x 3 Mostly, a function is what you learned it was all through f(x) = x mod 4 K-12 mathematics, with strange vocabulary to make it more interesting… f(x) = x A function f:A B maps values from its domain A to its co-domain B . Domain Co-domain f(x) = x 3 f(x) = x mod 4 f(x) = x Not every function is easy to plot! Unit 12: Functions Unit 12: Functions 3 5 1
What is a Function? What is a Function? A function f:A B maps values from its domain A to Not every function has to do with numbers… its co-domain B . A function f:A B maps values from its domain A to its co-domain B . f(control, data1, data2) = (~control data1) (control data2) Domain Co-domain f(x) = ~x f(x,y) = x y Domain? Co-domain? f(x) = x ’s phone # Unit 12: Functions 7 Unit 12: Functions 9 What is a Function? What is a Function? A function f:A B maps values from its domain A function f:A B maps values from its domain A to its A to its co-domain B . co-domain B . f can’t map one element of its domain to more than one Alan element of its co-domain: Steve 111 x A, y 1 ,y 2 B, 121 [(f(x) = y 1 ) (f(x) = y 2 )] (y 1 = y 2 ). Paul Patrice 211 f Karon Why insist on this? A B George Domain? Co-domain? Other examples? Unit 12: Functions Unit 12: Functions 11 13 2
Not a Function Function Terminology A function f:A B maps values from its domain A to its co-domain B . Why isn’t this a function? For f to be a function, it must map every element in its domain: x A, y B, f(x) = y. f Why insist on this? B A Warning: some mathematicians would say that makes f “total”. (The Laffer Curve: a non-functional tax policy.) Unit 12: Functions 15 Unit 12: Functions 17 Not a Function Function Terminology A function f:A B maps values from its domain A to its co-domain B . Anne f(x) is called the image of x (under f ). Alan x is called the pre-image of f(x) (under f ). Steve 111 Paul 121 f Patrice 211 Karon George x B A y Unit 12: Functions Unit 12: Functions 19 21 3
Trying out Terminology f(x) = x 2 What is the image of 16? PART 2 What is the range of f ? IN CLASS PAGES f(x) x Unit 12: Functions 23 Unit 12: Functions 25 Pre-Class Learning Goals Quiz 10 By the start of class, you should be able to: In General: Define the terms domain, co-domain, range, image, and pre- image Specific issues: Use appropriate function syntax to relate these terms (e.g., f : A → B indicates that f is a function mapping domain A to co-domain B). Determine whether f : A → B is a function given a definition for f as an equation or arrow diagram. Unit 12: Functions Unit 12: Functions 26 27 4
In-Class Learning Goals Outline By the end of this unit, you should be able to: Injective Functions Define the terms injective (one-to-one), surjective (onto), Surjective Functions bijective (one-to-one correspondence), and inverse. Determine whether a given function is injective, surjective, Bijective Functions and/or bijective. Determine whether the inverse of a given function is a Inverse Operations. function. Unit 12: Functions 28 Unit 12: Functions 29 Injective Functions Trying out Terminology f: R R 0 Some special types of functions: f(x) = x 2 A function f : A → B is injective (one-to-one) if Injective? ∀ x ∈ A, ∀ y ∈ A, x ≠ y → f(x) ≠ f(y). What if f: R 0 R 0 ? In the arrow diagram: at most one arrow points to each f(x) element of B. Not injective: f(George) = f(Steve) Injective CPSC 110/201 George CPSC 110 George CPSC 121/202 Steve CPSC 121 Steve CPSC 121/203 Gail CPSC 210 Neil CPSC 210/BCS Kimberly CPSC 310 Kimberly CPSC 310 x Neil CPSC 319 Gail CPSC 319 R 0 , Z 0 are the sets of non-negative real, integer numbers Unit 12: Functions Unit 12: Functions 30 31 5
Trying out Terminology Trying out Terminology f(x) = |x| (the absolute value of x ) f:{s|s is a 121 student} {A+, A, …, D, F} Injective? f(s ) = s’s mark in 121 a. Yes, if f: R R 0 b. Yes, if f: R 0 R c. Yes, for some other domain/co-domain Is f injective? d. No, not for any domain/co-domain a. Yes e. None of these is correct b. No f(x) c. Not enough information x Unit 12: Functions 32 Unit 12: Functions 33 Trying out Terminology Outline f:{s|s is a 121 student} {A+, A, …, D, Injective Functions F} Surjective Functions What if we didn’t know what f represented, only its “type” and the fact that there are 300 CPSC 121 Bijective Functions students: Inverse Operations. Is f injective? a. Yes b. No c. Not enough information Unit 12: Functions Unit 12: Functions 34 35 6
Surjective Functions Trying out Terminology A function f : A → B is surjective (onto) if f: R R 0 ∀ y ∈ B, ∃ x ∈ A, f(x) = y. f(x) = x 2 Can we define it in terms of range and co-domain? f(x) Surjective? In the arrow diagram: at least one arrow points to each element of B. What if f: R R ? What if f: Z Z 0 ? Surjective Not Surjective George CPSC 121/202 George Steve CPSC 121 Steve CPSC 121/203 Gail CPSC 210 x Neil CPSC 210/BCS Kimberly CPSC 310 Kimberly CPSC 310 Neil CPSC 319 CPSC 319 R 0 , Z 0 are the sets of non-negative real, integer numbers Unit 12: Functions 36 Unit 12: Functions 37 Trying out Terminology Trying out Terminology f:{s|s is a 121 student} {A+, A, …, D, F} f(x) = x Is f surjective? f(s ) = s’s mark in 121 f(x) a. Yes, for f: R R 0 ? b. Yes, for f: R 0 R ? Is f surjective? c. Yes, for f: R Z ? a. Yes d. No, not for any b. No domain/co-domain c. Not enough information x e. None of these Could we ever know that f was surjective just by knowing is correct f ’s domain and co -domain? Unit 12: Functions Unit 12: Functions 38 39 7
Surjective Functions So Far Hash Functions Which combinational circuits with one output are surjective? A hash function maps its input onto the indexes of an array so we can store arbitrary data in an array Every such circuit. a. e.g. h(x ) = x mod k b. Any such circuit that represents a contingency where k is the array size (neither a tautology nor contradiction). c. Only the ones equivalent to an inverter. If it’s not surjective , then we “waste” entries in the No such circuit is surjective. d. array that are never mapped to! None of these is correct. e. Unit 12: Functions 40 Unit 12: Functions 41 Outline Bijective Functions A function f : A → B is bijective (also one-to-one Injective Functions correspondence) if it is both one-to-one and onto (both Surjective Functions injective and surjective). In the arrow diagram: exactly one arrow points to each Bijective Functions element of B. Inverse Operations. Not Bijective either Not Bijective CPSC 110/201 George George CPSC 121/202 Steve CPSC 121 Steve CPSC 121/203 Gail CPSC 210 Neil CPSC 210/BCS Kimberly CPSC 310 Kimberly CPSC 310 Neil CPSC 319 CPSC 319 Unit 12: Functions Unit 12: Functions 42 43 8
Bijective Functions Trying out Terminology This is bijective f(x) = x 2 f:? ? Bijective f(x) Bijective for what Steve CPSC 121 domain/co-domain? Gail CPSC 210 Kimberly CPSC 310 Neil CPSC 319 x Unit 12: Functions 44 Unit 12: Functions 45 Outline Inverse of a Function The inverse of a function f: A → B, denoted f -1 , is Injective Functions f -1 :B A . Surjective Functions f -1 (y) = x f(x) = y . Bijective Functions In other words: Inverse Operations. If we think of a function as a list of pairs. E.g. f(x) = x 2 : { (1, 1), (2, 4), (3, 9), (4, 16), ... } Then f -1 is obtained by swapping the elements of each pair: f -1 = { (1, 1), (4, 2), (9, 3), (16, 4), ... } Unit 12: Functions Unit 12: Functions 46 47 9
Inverse of a Function Trying out Terminology Is f -1 a function? What’s the inverse of each of these f s? A. Yes, always. B. No, never. Alan Alan 121/202 111 George George C. Yes, but only if f is injective. 121/203 121 Paul Paul 121/BCS D. Yes, but only if f is surjective. Steve Steve 211 211/201 Karon Karon E. Yes, but only if f is bijective. 211/202 Patrice Patrice 211/BCS Can we prove it? Unit 12: Functions 48 Unit 12: Functions 49 Trying out Terminology Appendix 3: An Inverse Proof Theorem : If f : A B is bijective, then f(x) = x 2 f -1 : B A is a function. What’s the inverse of f ? Proof: We proceed by antecedent assumption. f(x) Assume f : A B is bijective. What should the Consider an arbitrary element y of B . domain/co-domain be? Because f is surjective, there is some x in A such that f(x) = y . Because f is injective, that is the only such x . f -1 (y) = x by definition; so, f -1 maps every x element of B to exactly one element of A . QED Unit 12: Functions Unit 12: Functions 50 53 10
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