1. CPSC 121: Models of Computation 2011 Winter Term 1
Functions
Steve Wolfman, based on notes by Patrice Belleville and others

2. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

3. Learning Goals: Pre-Class
By the start of class, you should be able to:
Define the terms domain, co-domain, range, image, and pre-image
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.

4. Learning Goals: In-Class
By the end of this unit, you should be able to:
Define the terms injective (one-to-one), surjective (onto), bijective (one-to-one correspondence), and inverse.
Determine whether a given function is injective, surjective, and/or bijective.
Apply your proof skills to proofs about the properties (e.g., injectiveness, surjectiveness, bijectiveness, and function-ness) of functions and their inverses.
Discuss point of learning goals.

5. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

6. What is this?
data1
output
data2
control

7. What is this?
data1
output
data2
control

8. What is this?
data1
output
data2
control

9. What is this?
In the lab, you implemented a multiplexer and then used it as a piece in larger circuits.
You abstracted from the concrete implementation to a description of its function:
f(control, data1, data2) = output = (~control  data1)  (control  data2) = data1 if control is 0, but data2 otherwise
control
data1
data2
output

10. Functions, Abstraction, and CPSC 121
Computer scientists use many abstraction levels, with reasoning and execution tools at each level.
In 121, we encounter abstraction levels like: wiring physical gates, computer-based hardware design techniques, propositional logic, predicate logic, sets, …
Functions are a model that let us talk about how computations work and fit together.
Bonus: functions alone are enough for all of computation, using the “λ calculus”.

11. Functions We’ve Used Any combinational logic circuit
Lists, as long as they’re immutable (unchangeable): A(i) = the ith value in A.
The Power Set operation (mapping a set to another set)
Addition, subtraction, multiplication, ...
Sequential circuits and mutable lists aren’t functions. Why not? (Actually we can model them as functions by making time explicit)

12. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

13. CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
What is a Function?
Mostly, a function is what you learned it was all through K-12 mathematics, with strange vocabulary to make it more interesting…
A function f:A  B maps values from its domain A to its co-domain B.
Domain Co-domain
f(x) = x3
f(x) = x mod 4
f(x) = x
Look, sets!

14. Plotting Functions f(x) = x3 f(x) = x mod 4 f(x) = x
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Plotting Functions
f(x) = x3
f(x) = x mod 4
f(x) = x
Not every function is easy to plot!

15. What is a Function? Not every function has to do with numbers…
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
What is a Function?
Not every function has to do with numbers…
A function f:A  B maps values from its domain A to its co-domain B.
Domain Co-domain
f(x) = ~x
f(x,y) = x  y
f(x) = x’s phone #

16. CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
What is a Function?
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?

17. CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
What is a Function?
A function f:A  B maps values from its domain A to its co-domain B.
Domain?
Co-domain?
Alan
Steve
111
Paul
121
Patrice
211
Karon
George
Other examples?

18. CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
What is a Function?
A function f:A  B maps values from its domain A to its co-domain B.
f can’t map one element of its domain to more than one element of its co-domain:
x  A, y1,y2  B, [(f(x) = y1)  (f(x) = y2)]  (y1 = y2).
Why insist on this? f A B

19. Why isn’t this a function?
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Not a Function
Why isn’t this a function?
(The Laffer Curve: a non-functional tax policy.)

20. Function Terminology f A B
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Function Terminology
A function f:A  B maps values from its domain A to its co-domain B.
For f to be a function, it must map every element in its domain:
x  A, y  B, f(x) = y.
Why insist on this? f A B
Warning: some mathematicians would say that makes f “total”.

21. Not a Function Anne Alan Steve 111 Paul 121 Patrice 211 Karon George
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Not a Function
Anne
Alan
Steve
111
Paul
121
Patrice
211
Karon
George
Foiled by sabbatical.

22. Function Terminology f A B
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Function Terminology
A function f:A  B maps values from its domain A to its co-domain B.
f(x) is called the image of x (under f).
x is called the pre-image of f(x) (under f). f A B x y

23. Trying out Terminology
CORRESPONDS TO TEXTBOOK READING (NOT COVERED IN CLASS)
Trying out Terminology
f(x) = x2
What is the image of 16?
What is the range of f?
f(x) x

24. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

25. Function Properties: Injective
A function f:A  B is injective (also one-to-one) if each image is associated with at most one pre-image: xA, yA, xy  f(x)  f(y).
Injective?
Injective?
111
Alan
Alan
121/202
Steve
Steve
111
121/203
Paul
121/BCS
Paul
121
Patrice
Patrice
211
211/201
Karon
Karon
211/202
George
George
211/BCS

26. Trying out Terminology
f(x) = x2
Injective?
What if f:R+  R+?
f(x) x

27. Trying out Terminology
f(x) = |x| (the absolute value of x)
Injective?
Yes, if f:R  R+
Yes, if f:R+  R
Yes, for some other domain/co-domain
No, not for any domain/co-domain
None of these is correct
f(x) x

28. Trying out Terminology
f:{s|s is a 121 student}  {A+, A, …, D, F}
f(s) = s’s mark in 121
Is f injective?
Yes No
Not enough information

29. Trying out Terminology
f:{s|s is a 121 student}  {A+, A, …, D, F}
f(s) = s’s mark in 121
What if we didn’t know what f represented, only its “type” and the fact that there are lots of 121 students:
Is f injective?
Yes No
Not enough information

30. Nifty Injective Function: Error-Detecting Codes
An error-detecting code encodes a normal number (or text) as a longer number (or text) in such a way that a small error in transcribing the encoded value will obviously not be a result of the code’s function (and furthermore, for error-correcting codes, will be “closer” to one particular valid code than any other).
If it were not injective, then applying it would conflate two of the inputs, causing exactly the sort of error/confusion that the code is intended to avoid!
Error-detecting codes must be injective. Why?

31. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

32. Function Properties: Surjective
f:A  B is surjective (also onto) if every element of the co-domain has a pre-image: y  B, x  A, y = f(x).
Surjective?
Surjective?
111
Alan
Alan
121/202
Steve
Steve
121/203
Paul
121
121/BCS
Paul
Patrice
Patrice
211
211/201
Karon
Karon
211/202
George
George
211/BCS
Can we define “surjective” in terms of “range” and “co-domain”?

33. Trying out Terminology
f(x) = x2
f:R  R0?
Surjective?
What if f:R  R?
What if f:Z  Z0?
f(x) x

34. Trying out Terminology
f(x) = x
Is f surjective?
Yes, for f:R  R0?
Yes, for f:R0  R?
Yes, for f:R  Z?
No, not for any domain/co-domain
None of these is correct
f(x) x

35. Trying out Terminology
f:{s|s is a 121 student}  {A+, A, …, D, F}
f(s) = s’s mark in 121
Is f surjective?
Yes No
Not enough information
Could we ever know that f was surjective just by knowing f’s domain and co-domain?

36. Trying out Terminology
Could we ever know that f was surjective just by knowing f’s domain and co-domain (and their cardinalities)?
Yes No
Not enough information

37. Nifty Surjective Function: Cryptographic Hash
A hash function maps its input onto the indexes of an array so we can store arbitrary data in an array. If it’s not surjective, then we “waste” entries in the array that are never mapped to!
Of course, if it’s not injective, then we have a different kind of trouble!
Old example: Lossy compression maps a file to another, hopefully smaller file. It’s “lossy” in the sense that the original can’t necessarily be uniquely extracted after compression (so, NOT injective). It’s handy if it’s surjective (or close to it), since if it is NOT surjective, then it “wastes” encodings, forcing the encodings that are actually used to be longer and decreasing the amount of compression.
“Good” cryptographic hashes should be but don’t have to be surjective… but they must not be injective.

38. Surjective Functions So Far
Which combinational circuits with one output are surjective?
Every such circuit.
Any such circuit that represents a contingency (neither a tautology nor contradiction).
Only the ones equivalent to an inverter.
No such circuit is surjective.
None of these is correct.

39. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

40. Function Properties: Bijective
A function f:A  B is bijective (also one-to-one correspondence) if it is both one-to-one and onto (both injective and surjective).
Every element in the domain has exactly one unique image. Every element in the co-domain has exactly one unique pre-image.

41. Function Properties: Bijective
A function f:A  B is bijective (also one-to-one correspondence) if it is both one-to-one and onto.
Bijective?
Bijective?
Alan
Alan
121/202
111
Steve
Steve
121/203
Paul
Paul
121
121/BCS
Patrice
Patrice
211
211/201
Karon
Karon
211/202
George
George
211/BCS

42. Trying out Terminology
f(x) = x2
f:?  ?
Bijective for what domain/co-domain?
f(x) x

43. Nifty Bijective Function: Encryption/Lossless Compression
What it says is that if lossless compression actually compresses some of the time, there must be other cases where it fails to compress (since it’s mapping bijectively from one domain (legal files) to the same domain). Try compressing things like executables, JPEG images, and already-compressed files, and you’ll see what we mean! (these compress poorly and sometimes actually expand.. Although many compression programs will simply not perform the compression if the file would expand.)
Two sets have the same cardinality if we can put them in a bijection. What does that say about lossless compression?

44. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

45. Function Operations: Inverse
The inverse of a function f:A  B is f-1:B  A. f(x) = y  f-1(y) = x.
How can we tell whether f-1 is a function?
(Hint: what would make f-1 not a function?)
Can we prove it?

46. Trying out Terminology
What’s the inverse of each of these fs?
Alan
Alan
121/202
111
Steve
Steve
121/203
Paul
121
121/BCS
Paul
Patrice
Patrice
211
211/201
Karon
Karon
211/202
George
George
211/BCS

47. Trying out Terminology
f(x) = x2
What’s the inverse of f?
What should the domain/co-domain be?
f(x) x

48. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

49. Proving a function injective
Recall: A function f : A  B is injective exactly when: xA, yA, xy  f(x)  f(y). A typical approach is to prove the contrapositive by antecedent assumption: assume f(x) = f(y) and show that x = y.

50. Problem: x3 + 5 is injective
Theorem: f(x) = x3 + 5 is injective, where f : Z  Z.

51. Proving a function surjective
Recall: A function f : A  B is surjective exactly when: y  B, x  A, y = f(x). That existential gives us a lot of freedom to pick a witness! A typical approach is to “solve for” the necessary x given a y.

52. Problem: w + 2z is surjective
Theorem: f(w,z) = w + 2z is surjective, where f : (Z  Z)  Z.

53. Proving a function bijective
Prove that it’s injective. Prove that it’s surjective. Done.

54. Proving a function has an inverse
Prove that it’s bijective. Done.

55. An Inverse Proof
Theorem: If f:A  B is bijective, then f-1 : B  A is a function.
Recall that f-1(f(x)) = x.

56. Aside: Functions are Just Sets
We can define functions in terms of sets and tuples (and we can define tuples in terms of sets!).
The function f:A  B is the set {(x, f(x)) | x  A}.
f is a subset of A  B!

57. Outline Prereqs, Learning Goals, and Quiz Notes
Functions as “Computation Abstractions”
Terminology (out-of-class)
Properties: Injective, Surjective, Bijective
Inverse Operation
Function Proofs
Next Lecture Notes

58. Learning Goals: In-Class
By the end of this unit, you should be able to:
Define the terms injective (one-to-one), surjective (onto), bijective (one-to-one correspondence), and inverse.
Determine whether a given function is injective, surjective, and/or bijective.
Apply your proof skills to proofs about the properties (e.g., injectiveness, surjectiveness, bijectiveness, and function-ness) of functions and their inverses.
Discuss point of learning goals.

59. Learning Goals: “Pre-Class” (REPEAT)
The pre-class goals are to be able to:
Trace the operation of a deterministic finite-state automaton (represented as a diagram) on an input, including indicating whether the DFA accepts or rejects the input.
Deduce the language accepted by a simple DFA after working through multiple example inputs.

60. Extra Slides…