Lecture 04: SETS AND FUNCTIONS 1.6, 1.7, 1.8

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Lecture 04 SETS AND FUNCTIONS1.6, 1.7, 1.8
CS1050 Understanding and Constructing Proofs
Spring 2006

• Jarek Rossignac

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Lecture Objectives

• Sets, specification, notation, construction
• Power sets, ordered pairs, relations
• Operations on sets and pointsets
• Functions and composition

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What is a set?

• Unordered collection of elements (members)
• A set contains its members
• We may define a set by listing its elements
• S a,b,c
• S1,2,3
• or by a property (set builder notation)
• S a a?Z ? (?k?Z a2k1)
• S x x is a positive integer less than 200
  and divisible by 3

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What are the important sets of numbers?

• R real numbers
• Q p/q p?Z, q?Z, q?0
• Z 1, 0, 1, . integers
• N 0,1,2. natural numbers
• Z 1,2,3 . positive integers

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When are two sets equal?

• When they have the same elements
• a,b,c a,c,b a,b,c,b,b

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What is a Venn diagram?

• Graphic depiction of (topological) set
  relationship

Z
N
0
Z
Q
R
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What operators/notation are used for sets?

• Sets in upper case. Elements in lower case.
• s?S s is an element of S. s?S s is not in S
• S number of elements (cardinality) of a finite
  set S
• ST S equals T (equality)
• ?S elements not in S (complement)
• S?T elements in S and T (intersection)
• S?T elements in S or T (union)
• ST S??T, elements of S that are not in T
  (difference)
• S?T S?T S?T (symmetric difference)
• S ? T S included in or equal to T (inclusion)
• S ? T S included in (but not equal to) T ( proper
  inclusion)
• ? empty (i.e. null) set, also denoted .
  Different from ?

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What are the subsets of a set S?

• All sets whose elements are all in S
• This always includes
• The empty set, ?, since ? ? S
• The whole set, S , since S ? S

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What is the power set of set S?

• The power set P(S) of set S is the set of all
  subsets of S.
• P(a,b) ?, a,b,a,b

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What is the power set of ??

• P(?) ?
• Remember that ? is a set, not an element

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What is the power set of ??

• P(?) ?, ?
• Remember that ? has one element, which is a set
• Hence its power set will contain two elements
• the empty set ?
• the set itself ?

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What is an ordered n-tuple?

• The ordered n-tuple (a,b,c) is the ordered list
  of elements
• a, b, c
• Two n-tuples are equal , (a,b,c) (d,e,f) ,
• when corresponding elements are equal, (ad, be,
  cf).

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What is an ordered pair?

• An n-tuple of two elements
• For example,
• The (x,y) coordinates of a point in the plane
• The vertices (a,b) of an oriented edge or link in
  the graph

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What is the Cartesian product of two sets?

• A ? B (a,b) a?A, b?B
• set of ordered pairs (a,b) such that a is an
  element of A and b is an element of B
• a,b ? c,d (a,c), (a,d), (b,c), (b,d)
• Note that in general, A ? B ?B ? A

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How to use quantifiers with sets?

• ?s?S means for all s in S
• ?s?S means there exists an element s in S
• ?!s?S means there exists a single element s in S

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What are unions and intersections of sets?

• Union S?T u u?S ? u?T
• Intersection S?T u u?S ? u?T

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What are the differences of sets?

• Complement ?S u u?S, also denoted !S
• Implicitly defined in a universal set U as ?S
  US
• Difference ST S??T u u?S ? u?T
• Symmetric difference S?T (ST) ? (TS)

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Individual exercise

• Prove that (S?T) (S?T) (ST) ? (TS)

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When are two sets disjoint?

• When their intersection is empty S?T ?
• We often have to prove that two or more sets are
  pairwise disjoint
• Often it is necessary to test whether two sets
  are disjoint
• Components of a car engine do not interfere
• Objects do not collide in an animation

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What is S?T ?

• The cardinality, S?T, is the number of elements
  of the union, S?T, of the two finite sets S and
  T.
• S?T S T S?T
• we need to subtract elements of S?T, since they
  were counted both in S and in T.

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Identities of Boolean operators

• Identity A?UA, A?UA
• Idempotence A?AA, A?AA
• Domination A?UU, A???
• Complement A??A U, A??A?
• Commutativity A?BB?A, A?BB?A,
• Associativity A?(B?C)(A?B)?C, A?(B?C)(A?B)?C
• Distributivity A?(B?C)(A?B)?(B?C),
  A?(B?C)(A?B)?(B?C)
• !S denotes ?S Sand has higher priority than ? , ?
  , and
• Complementation !(!A)A
• de Morgan !(A?B) !A?!B , !(A?B) !A?!B
• Absorption A?(A?B)A, A?(A?B)A

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Relation between sets and Booleans?

• Define the propositions A(u)(u?A) and
  B(u)(u?B).
• Then
• A?B u ? U A(u)?B(u)
• A?B u ? U A(u)?B(u)
• !A u ? U A(u)
• Hence, there is a correspondence between
• ? and ? (AND, )
• ? and ? (OR, )
• ! and (AND NOT, !)
• ? and F (false)
• U and T (true)
• Translate the previous identities into logic

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Prove !(A?B) !A?!B using inclusion

• Hint To prove that S T, prove that (S?T) ?
  (T?S)
• To prove that !(A?B) ? !A ? !B
• assume that for some arbitrary x, x ? !(A?B)
• hence (x?A ? x?B)
• hence x?A ? x?B
• hence x?!A ? x?!B
• it follows that x ? (!A ? !B)
• Since our choice of x?!(A?B) was arbitrary,
  ?x?!(A?B) x?(!A ? !B)
• Hence!(A?B) ? !A ? !B
• To prove that !A ? !B ? !(A?B)
• prove that it follows that x ? !(A?B)

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Prove !(A?B) !A?!B using builder

• This time we use the set builder notation
• !(A?B) x x ? A?B
• x (x ? A?B)
• x ((x ? A) ? (x ? B))
• x (x ? A) ? (x ? B)) de Morgan
• x x ? !A ? x ? !B
• x x ? (!A?!B)

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Prove A?(B?C)(A?B)?(A?C) using cells

• Test one element in each cell against both
  expressions

A
B
C
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Exercise

• Provide a concise Boolean expression for the set
  marked by the green cells

A
B
C
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Prove !(A?B) !A?!B using a truth table

• Test all combinations

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How to represent finite sets?

• Consider a finite domain Ua,b,c,d. where
  Ult220
• We can represent any subset of U by listing its
  elements.
• But working on such a representation may be
  expensive.
• A more efficient representation may be derived
  if we assume an ordering of the elements.
• For example the n-tuple (a,b,c,d)
• Represent a subset S of U by a classification
  vector (bit string)
• V(S) 0110 which indicates that
• a?S, b?S, c?S, d?S

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How to perform Boolean operations on sets?

• Consider 2D arrays Aij and Bij of bits
  representing digitized versions of 2D sets A and
  B
• (Pixel(i,j)?A) ? (Aij 1) red if 1, green
  if 0
• (Pixel(i,j)?B) ? (Bij 1)
• How to represent and compute A?B?

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How to deal with continuous pointsets?

• Use a constructive model SA?(B?C)
• Provide point/primitive classification methods
• A.isIn(x,y) returns true if point(x,y) ? A
• For example, if A is rect(lx,ly,hx,hy), then
• A.isIn(x, y) return((lxltx)(xlthx)
  (lylty)(ylthy))
• Provide Boolean classification methods
• U.isIn(L,R) combines truth values L and R
• For example, if U is a union operator, then
• U.isIn(L,R) return ( L R )

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What is Constructive Solid Geometry?

• A CSG model is a binary tree representing a 3D
  shape
• Leaves represent solid primitives (blocks,
  spheres, bunnies)
• Nodes represent Boolean operators (?, ?, )

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What is a relation from set A to set B?

• It is a subset of A ? B
• Think of it as links from elements of A to
  elements of B
• A relation assigns to each set of A zero or more
  elements of B

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What is a function from A to B

• A relation assigning exactly one element of B to
  each element of A.
• Note that it may assign the same element b?B to
  two different elements of A.
• We write it f A ? B (here ? is not an
  implication)
• ?a?A ?b?B bf(a)
• A is the domain of f

A
B
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What is the image of S by f?

• b?B is the image of all elements a?A for which
  bf(a)
• The image f(S) of a subset S of A is the set of
  the images of elements of S.

S
A
B
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What is the range of a function?

• The image of A.
• (The set of images of elements of A).

A
B
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When is a function injective?

• f is injective (one-to-one) when f(x)f(y) ?
  xy

Not injective
Injective
A
A
B
B
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When is a function surjective?

• f is surjective (onto) when ?b?B ?a?A f(a)b

Not surjective
Surjective
A
A
B
B
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When is a function bijective?

• f is bijective when it is both injective and
  surjective
• (one-to-one and onto)
• one-to-one correspondence
• ?b?B ?!a?A f(a)b

Not bijective
Bijective
A
A
B
B
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What is the inverse of a function?

• The inverse is only defined for bijective
  functions
• The inverse f1 of f maps each b?B to a?A where
  bf(a)
• f1(b)a ? bf(a)

f
f1
A
A
B
B
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What is the composition of two functions?

• Let f A ? B and g B ? C
• The composition f?g A ? C of f and g is defined
  as
• f?g(a) f(g(a))

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Application to computer graphics

• Translation
• Rotation
• Composition
• Do rotations and translations commute?

T(2,1)
R(30)
T(2,1)?R(30)
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What is the graph of a function?

• Set of ordered pairs (a,b) a?A, b?B, bf(a)
• Sometimes we can plot it
• Here f maps points (x,y) on the plane into height

f(x,y) x22y41
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What are floor and ceiling?

• The floor ?x? of x is the larges integer not
  exceeding to x
• The ceiling ?x? is the smallest integer x
• Properties
• x1lt ?x? x ?x? lt x1
• ?x??x?, ?x? ?x?
• ?xn??x?n, ?xn? ?x?n
• Compute ?0.5? and ?0.5?
• How many bytes (8-bit strings) do you need to
  encode 100 bits?
• Prove or disprove ?xy? ?x? ?y?

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Quick test

• Let A and B be two sets
• What is the symmetric difference of A and B?
• When are A and B disjoint?
• How would you prove that AB?
• What is the cardinality of A?B?
• What is Constructive Solid Geometry
• Let f be a function from A to B
• What property does f have?
• What is the image of a subset S of A by f?
• When is f injective?
• When is the inverse of f defined?
• What is the floor of 3.2?

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Assigned Reading

• Sections 1.6, 1.7, 1.8

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Assigned Exercises (for quiz)

• 1.6 p 85
• 18
• 1.7 p 95
• 11, 15, 16, 21, 27, 28, 29, 30, 31, 32 ,33
• 1.8 p 109-110
• 28, 32, 35.b), 38, 40

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Assigned Project

• P1 is due Tu Jan 24 (on your PPP)
• Email TA only if
• you did not submit P0.
• you plan to have a late submission.
• you changed your PPP url.