CPSC 121: Models of Computation Unit 12 Sets and Functions Based - - PowerPoint PPT Presentation

Based on slides by Patrice Belleville and Steve Wolfman

CPSC 121: Models of Computation

Unit 12 Sets and Functions

PART 1

SETS

Unit 12: Sets and Functions 2

What is a Set?

A set is an unordered collection of objects. The objects in a set are called members.

(a  S indicates a is a member of S; a  S indicates a is not a member of S)

A set contains its members.

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Unit 12: Sets and Functions

Describing Sets

Some sets…

A = {1, 3, 9} B = {1, 3, 9, 27, snow} C = {1, 1, 3, 3, 9, 9}

Some sets, using set builder notation: A = {x  N | y  N, x = 5y} B = {2i - 1 | i is a prime} C = {n  Z | 0 < n  100}

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Unit 12: Sets and Functions

Containment

A set A is a subset of a set B iff x  U, x  A  x  B. We write A is a subset of B as A  B. If A  B, can B have elements that are not elements of A? Yes, but A can’t have elements that are not elements of B.

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Unit 12: Sets and Functions

Set Equality

Sets A and B are equal — denoted A = B — if and only if x  U, x  A  x  B. Can we prove that that’s equivalent to A  B and B  A? Yes, using a standard predicate logic proof in which we note that p  q is logically equivalent to p  q  p  q.

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Unit 12: Sets and Functions

Set Union

The union of A and B — denoted A  B — is {x  U | x  A  x  B}. A  B is the blue region...

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A B

U

Unit 12: Sets and Functions

Set Intersection

The intersection of A and B — denoted A  B — is {x  U | x  A  x  B}. A  B is the dark blue region...

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A B

U

Unit 12: Sets and Functions

Set Difference

The difference of A and B — denoted A - B — is {x  U | x  A  x  B}. A – B is the pure blue region.

U

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A B

U

Unit 12: Sets and Functions

Set Complement

The complement of A — denoted A — is {x  U | x  A}. A is everything but the blue region.

U Can we express this as a set difference?

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A

U

Unit 12: Sets and Functions

Historical Notes on Sets

 Mathematicians formalized set theory to create a

foundation for all of mathematics. Essentially all mathematical constructs can be defined in terms of sets.

 Hence sets are a powerful means of formalizing new

ideas.

 But we have to be careful how we use them!

Unit 12: Sets and Functions 11

Russell's Paradox

 At the beginning of the 20th century Bertrand Russell

discovered inconsistencies with the "naïve" set theory.

 Let R be the set of all sets that

do not contain themselves. That is R = { x | ~xϵx }.

 Then:

R  R  ~R  R. which is a contradiction.

 Set theory has been restricted in a way that disallow

this kind of sets.

Unit 12: Sets and Functions 12

Same question, different form: “Imagine a barber that shaves every man in town who does not shave himself. Does the barber shave himself?”

Set Cardinality

 Cardinality: the number of elements of a set S,

denoted by |S|.

 The cardinality of the set:

{ 1, 2, 3, { a, b, c }, snow, rain } is 6.

Unit 12: Sets and Functions 13

Power Sets

 The power set of a set S, denoted P (S), is the set

whose elements are all the subsets of S. That is P (S) = { T | T  S}

 Given the definitions

A = { a, b, f }, B = { b, c }, then:

P (B) = {, {b}, {c}, {b, c} } P (A - B) = { , {a}, {f}, {a, f} }

 Theorem :

If S is a finite set then |P(S)| = 2|S|

Unit 12: Sets and Functions 14

Tuples

 An ordered tuple (or just tuple) is an ordered

collection of elements. (An n-tuple is a tuple with n elements.)

 Two tuples are equal when their corresponding

elements are equal.

 Example:

(a, 1, ) = (a, 5 – 4, A  A) (a, b, c)  (a, c, b)

15 Unit 12: Sets and Functions

Cartesian Product

 The cartesian product of two sets S and T, denoted

S x T, is the set of all tuples whose first element is drawn from S and whose second element is drawn from T

 In other words,

S x T = { (s, t) | s ∈ S  t ∈ T }.

• Each element of S x T is called a 2-tuple or a pair.

Unit 12: Sets and Functions 16

Calculating Cartesian Products

 What is {a,b}  {1,2,3}:

1 2 3 a b ( a , 3 ) ( a , 2 ) ( a , 1 ) ( b , 3 ) ( b , 2 ) ( b , 1 ) The order of the tuples definitely matters!

17 Unit 12: Sets and Functions

PART 2

FUNCTIONS

Unit 12: Sets and Functions 18

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

R or Z or... R or Z or...

f(x) = x mod 4

Z or Z+ or... Z or Z+ or...

f(x) = x

R or R+ or... Z or Z+ or...

Unit 12: Sets and Functions 19

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 {T, F} {T, F} f(x,y) = x  y {T,F}{T,F} {T, F} f(x) = x’s phone # Set of people? 10-dig #s?

Unit 12: Sets and Functions 20

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).

A B

f

Unit 12: Sets and Functions 21

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.

Warning: some mathematicians would say that makes f “total”.

A B f

Unit 12: Sets and Functions 22

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).  The range of f is the set of all images of elements of

f’s domain. In other words: { f(x) | x  A }

A B f

x y

Unit 12: Sets and Functions 23

Injective Functions

Some special types of functions:

 A function f : A → B is injective (one-to-one) if

∀x ∈ A, ∀y ∈ A, x ≠ y → f(x) ≠ f(y).

 In the arrow diagram: at most one arrow points to each

element of B.

Unit 12: Sets and Functions

George CPSC 121/203 Ryan CPSC 121/202 Neil CPSC 210/BCS Kimberly CPSC 310 Gail CPSC 319 CPSC 110/201 George CPSC 110 Ryan CPSC 121 Gail CPSC 210 Kimberly CPSC 310 Neil CPSC 319 Injective Not injective: f(George) = f(Ryan)

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Trying out Terminology

f(x) = |x| (the absolute value of x) Is it injective?

• a. No, if f:R  R0
• b. Yes, if f:R0  R

Note: R0 = { x  R | x  0 }

x f(x)

Unit 12: Sets and Functions 25

Surjective Functions

 A function f : A → B is surjective (onto) if

∀y ∈ B, ∃x ∈ A, f(x) = y. Can we define it in terms of range and co-domain?

 In the arrow diagram: at least one arrow points to each

element of B.

Unit 12: Sets and Functions

George CPSC 121/203 Ryan CPSC 121/202 Neil CPSC 210/BCS Kimberly CPSC 310 CPSC 319 George Ryan CPSC 121 Gail CPSC 210 Kimberly CPSC 310 Neil CPSC 319 Not Surjective Surjective

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Trying out Terminology

f:R  R0 f(x) = x2 Surjective? Yes What if f:R  R? No What if f:Z  Z0? Yes x f(x)

R0, Z0 are the sets of non-negative real, integer numbers

Unit 12: Sets and Functions 27

Bijective Functions

 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).

 In the arrow diagram: exactly one arrow points to each

element of B.

Unit 12: Sets and Functions

George CPSC 121/202 Ryan CPSC 121/202 Neil CPSC 210/BCS Kimberly CPSC 310 CPSC 319 CPSC 110/201 George Ryan CPSC 121 Gail CPSC 210 Kimberly CPSC 310 Neil CPSC 319 Not Bijective Not Bijective either

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Bijective Functions

 This is bijective

Unit 12: Sets and Functions

Ryan CPSC 121 Gail CPSC 210 Kimberly CPSC 310 Neil CPSC 319 Bijective

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Trying out Terminology

f(x) = x2 f:?  ? Bijective for what domain/co-domain? x f(x)

Unit 12: Sets and Functions 30

Inverse of a Function

 The inverse of a function f: A → B, denoted f -1, is

f-1:B  A. f-1(y) = x  f(x) = y .

 In other words:

• If we think of a function as a list of pairs.

E.g. f(x) = x2 : { (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), ... }

 f -1 is a function only when f is bijective.

Unit 12: Sets and Functions 31

Appendix 3: An Inverse Proof

 Theorem: If f : A  B is bijective, then

f-1 : B  A is a function.

 Proof: We proceed by antecedent assumption.

• Assume f : A  B is bijective.
• Consider an arbitrary element y of B.

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

element of B to exactly one element of A. QED

Unit 12: Sets and Functions 34