Sets, Functions and Relations

1
Sets, Functions and Relations

• CSC 2110 Tutorial

2
Self introduction

• You can call me Isaac
• Im responsible for the tutorials of the first 3
  weeks and the first classwork
• If you have questions, you may email me at
  wsfung_at_cse.cuhk.edu.hk
• Or come to SHB 115

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What is a set?

• Q Give me one element of each of the following
  sets?
• The set of English letters,
• the set of English words that starts with d
  and ends with e,
• the set of all natural numbers,
• the set of all 8 digit telephone numbers,
• the set of factors of 30030,
• the set of integers x N100ltxlt120,
  
• The order of elements does not matter
• e.g. a, b, c is the same set as b, a, c
• Q If A (Jennifer, Ken, John, May) is a
  sequence of people who are ordered by their ages,
  is A just the same as the set Jennifer, Ken,
  John, May?
• The set 1, 2, 3, 3 is same as the set 1, 2, 3
• Q If I want to record the number of times my
  friends visit my home, can I do this by just
  adding his/her name into a set every time he/she
  visits me?

4
What can be in a set?

• A set may contain infinitely many elements
• e.g. the set of real numbers
• Q Give me another set that has infinitely many
  elements
• A set can also contain zero element
• e.g. the empty set,
• Q Can you give me another set that has zero
  elements?
• Types of elements doesnt matter,
• e.g. S 11/13, red, CSC2110, (10,10)
• A set can also be an element of some other set
• e.g. X 1, 1,2, 1,2,3
• Suppose A1, B2, C3
• Q Is A, B, C the same as 1, 2, 3?
• How many elements are there in the set 1,
  2 ?
• What about A A X?

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How can we specify a set?

• We can specify a set by
• listing all the members of the set, e.g. 1, 2,
  3
• Q Could you list all the elements of the set of
  integers?
• stating the properties of the set members,
• e.g. X x Z x is even
• Q Try specifying the sets of Fibonacci numbers
• (take home exercise)
• the results of set operations on some other sets
• e.g. A is the set of all quadrilaterals whose
  four sides have equal length (rhombus), B is the
  set of quadrilaterals which have two adjacent
  angles equal to (trapezium), C is the set of
  quadrilaterals such that the 2 pairs of opposite
  sides are parallel, so what is ,
  and ?

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How can we specify a set?

• Q What is the complement of the positive even
  integers?
• (if the universe is 1. positive integers, 2.
  even integers, 3. integers)
• Q If X has been defined to be the set of
  right-angled triangles and Y is the set of
  isosceles triangles, you are asked to specify the
  set of all right-angled isosceles triangles,
  which method would you prefer to use?
• Q Suppose Ma,b,c,d,e, Nb,d, Pc,e,
  Qb,c
• Express a,e in terms of these 4 sets using
  only basic set operations

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Venn Diagram

• When we work with just 2 or 3 sets, it is often
  useful to draw the Venn diagram
• Suppose the blue circle represents a set A and
  the red circle represents a set B
• Try to find the regions corresponding to the
  complement of A, A B,
• A B, A\B
• Try to derive the De Morgans law
• and

8
Venn Diagram, continued

• Suppose the red circle represents the set of
  multiples of 4, the blue circle represents the
  multiples of 15 and the yellow circle represents
  the multiples of 10.
• Try to figure out the meaning of each region
• Try to derive the distributive laws
• ,

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Subsets of a set

• e.g. the set of prime numbers is a subset of the
  natural numbers,
• the set of core courses is a subset of all the
  courses,
• a set is a subset of itself,
• the empty set is a subset of any set,
• the intersection of two sets A and B is always a
  subset of A and B,
• A and B are always subsets of the union of A and
  B
• Q Is the set of even numbers a subset of the
  composite numbers?
• Q Let x and y be two integers. If F is the set
  of factors of the largest common factor of x and
  y, is F a subset of the union of the set of
  factors of x and the set of factor of y?
• Q Let A1,2,3,4,5,6, B1,2,3,4,6,
  C1,3,4,5,6, D3,4
• Give me a subset of A that is not a subset of B
  and C but not contains D as its subset

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Subsets of a set, cont.

• X Y (X and Y contain the same elements)
• if and only if X Y and X Y (can you see
  why?)
• e.g. the set of multiples of 10 equals the
  intersection of the set of multiples of 2 and 5
• Q Let x and y be two integers
• If s is the smallest common multiples of x and
  y, does the set of factors of s equal the union
  of the set of factors of x and the set of factors
  of y?
• The power set of a set X, Pow(X) is the set of
  all the subsets of X
• e.g. Let X1, 2, 3. Pow(x) ,1, 2,
  3, 1,2, 2,3, 3,1, 1,2,3
• Q Give me the power set of the power set of 0,
  1
• Q Give me a set whose power set has only one
  element

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What is a function?

• e.g. the identity function, f(x)x
• e.g. the set membership function of a set X,
• e.g. is a function whose domain and
  codomain are sets of functions
• e.g. let x be a student ID,
• f(x)name of the student who has this student
  ID
• e.g. currency conversion formula,
• suppose x is the price of something in HK,
• f(x)the value of x in US

f
image of x
x
domain
codomain
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What is not a function?
g

• e.g. g(x)1/x is not a total function if the
  domain is , as 1/0 is undefined
• Q Is f(x)log(x) a total function if the
  domain is the set of real numbers larger than 0?
• e.g. Define f(x)y if y2x
• f(x) is not a function as 22(-2)24,
• the element 4 has two images 2 and -2 under f
• e.g. Let X be a set,
• f(X)an element of X,
• f(X) is not a functon as X may have no elements
  or X can have more than one elements

x
What is g(x)?
domain
codomain
X is not mapped to some element in the codomain
h
y
x
h(x)y or h(x)z?
z
domain
codomain
X is mapped to two elements in the codomain
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Surjective functions

• Roughly speaking, if a function is surjective,
  then each element in the codomain will have AT
  LEAST one arrow pointing to it
• e.g. f(x)sin(x), domain , codomain
  -1, 1
• f(x) is surjective
• Q Is f(x) still surjective if the codomain is
  ?
• e.g. f(x)1, domain , codomain 1
• f(x) is surjective but it is not surjective if
  we add anything other than 1 to its codomain
• e.g. Suppose f(x)x1 and the codomain is the
  set of even numbers
• Q If f(x) is surjective, what should be the
  domain of f(x)
• Q If there are more elements in the codomain
  than in the domain, can this function be
  surjective?

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Injective functions

• If a function is injective, then each element in
  the codomain can have AT MOST one arrow pointing
  to it
• e.g. f(x)course code of course x, domainset
  of courses,
• codomainset of course code
• f(x) is injective as no two courses share one
  course code
• e.g. f(x)cos(x), codomain-1, 1
• f(x) is not injective if the domain is
• but f(x) is injective if the domain is
• Q If a function is injective, can it be true
  that there are more elements in the domain than
  in the codomain?

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Bijection and inverse

• A function f is a bijection if it is total,
  surjective and injective
• e.g. f(x)x1 is a bijection between the set of
  even numbers and the set of odd numbers
• e.g. f(x)-x is a bijection between the set of
  positive numbers and the set of negative numbers
• If there is a bijection between 2 sets A and B,
  the sizes of A and B are the same
• e.g. We can construct a bijection between the set
  of English letters and the set 1,2,,26 to
  count the number of letters
• e.g. We can construct a bijection between the set
  of natural numbers and the set of rational
  numbers to count the size of a infinite set
• Q Try constructing a bijection between the set
  of natural numbers and the set of positive
  rational numbers
• If we reverse the direction of the arrows in a
  bijection, we get a new function, which is called
  the inverse of the original function.
• Q What is the inverse of f(x)(x-2)3?
• Q Does f(x)(x-2)2 have an inverse?

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Composite functions

• A composite function is a function formed by
  cascading 2 functions
• e.g. f(x) (sin(x))2 can be viewed as the
  composite of two functions h(y) and g(x) where
  h(y) y2 and g(x) sin(x)
• When we write ,
• actually we means
• When we want to evaluate f(x) we just pass the
  input x to g and then pass the output of g as the
  input of h, and finally we return the output of h
  as the output of f

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Composite functions, cont.
h
g
Q Is f(x) a total function if both g(x) and h(y)
are total functions? Assume f(x), g(x), h(y) are
all total functions. Is f(x) bijective if both
g(x) and h(x) are bijective? Are both g(x) and
h(x) bijective if f(x) is bijective?

• Graphically, we may join the graphs of the
  functions g and h to form the graph representing f

f
g
h
x
g(x)
f(x)h(g(x))
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Functions vs Relations

• In a function, each element in the domain is
  associated with one element in the codomain
• What if we want to associate each student with
  the course he/she has taken? A student may have
  taken gt1 course
• One approach is to model this by a function whose
  domain is the set of students and the codomain is
  the set of all possible combinations of courses
  (notice that the set of combinations of courses
  can be much larger than the set of all courses)
• Besides, the elements in the codomain are sets of
  courses. However what we want to model is the
  relationship between students and courses instead
  of relationship between students and set of
  courses

1130, 1500, 2100
Student A
2100, 2510, 3150
Student B
Student C
1500, 3150, 3160
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Functions vs Relations, cont.

• It is more natural to associate the students with
  the courses they take
• This requires us to allow each student to be
  associated with more than one courses
• We call such a mapping a relation

Using this approach, it is much easier to answer
questions like Who have taken 2100? Are there
any courses taken by both students A and B? Are
there any students who have taken both 2100 and
3150?
1130
Student A
1500
2100
Student B
2510
3150
Student C
3160
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Some special relations

• Here we only consider binary relations -
  relations between two objects
• You should have seen many binary relations before
  and many of them describe relations between 2
  elements of the same set
• e.g. a b, a lt b, equal to and smaller than
  are relations between pairs of real numbers
• e.g. P ? Q if and only if are relation between
  pairs of propositions
• e.g. Alice is a relative of Bob is a relation
  between two people
• e.g. John is a friend of Mary is also a
  relation between two people

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Graphs of relations

• When a binary relation is defined between
  elements of the same set, we can use another type
  of diagram to represent this relation
• In this type of diagram, we have only one set of
  points representing elements of the set. If two
  elements (x, y) are in the relation, we draw an
  arrow pointing from x to y (notice that the order
  matters, e.g. 2gt1, the converse 1gt2 is not true)
• e.g.

1
2
The is a friend of relation among some people
The defeats relation among some football teams
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More examples
3
4
5
Each point represents a person, and the arrows
corresponds to the has the same surname relation
Can you observe some of the properties of
diagrams 2, 3, 4? Can you tell what do they have
in common with diagram 5?
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Equivalence relations

• Notice that the elements in the diagram are
  divided into some disjoint subsets. Elements in
  the same subset have arrows pointing to each
  other (and themselves) but there are no edges
  crossing from one subset to another subset
• There are many relations which have diagrams
  similar to the diagram in the last example
• e.g. The is similar to relation on triangles,
  all equilateral triangles are similar
• e.g. The equals relation on rational numbers,
  2/3 4/6
• e.g. The has the same remainder when divided by
  7 relation on integers, 3 mod 7 10 mod 7
• Q How many disjoint subsets are there in this
  relation?
• We call these relations Equivalence relations

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Partition of a set

• If two sets A and B do not share any common
  elements, i.e. , we say that
  they are disjoint
• Suppose X1, X2, , Xn are subsets of a set X. If
• Their union is equal to X, and
• Every pair of them are disjoint
• Then we say that X1, X2, , Xn form a partition
  of X
• Refer back to the diagram in example 5. If the
  relation is a equivalence relation, we can form a
  partition by the following procedures
• Let each element form a subset which contains
  only this element
• Whenever there is an arrow pointing from an
  element x to an element y, combine the subset
  containing x and the subset containing y
• Continue until there is no arrow crossing two
  subsets
• The resulting collection of subsets is a
  partition of the set
• This partition has the properties highlighted in
  the last slide
• We call a subset in this partition, an
  equivalence class
• e.g. The even and odd numbers form two
  equivalence classes
• under the relation having the same remainder
  when divided by 2

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Symmetry, Transitivity and Reflexivity

• You may observe that there are some properties
  that are shared by example 5 and examples 2, 3
  and 4
• In examples 2 and 5, whenever there is an arrow
  pointing from an element x to an element y, then
  there is an arrow pointing from y to x.
• We say that such a relation is a symmetric
  relation
• In examples 3 and 5, for any three elements x, y
  and z, whenever there is an arrow pointing from x
  to y and an arrow from y to z, then there must be
  an arrow from x to z
• We say that this relation is a transitive
  relation
• In examples 4 and 5, every element in the set has
  an arrow pointing from itself to itself
• We say that this relation is a reflexive
  relation
• In fact, a relation is a equivalence relation if
  and only if it is symmetric, transitive and
  reflexive

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More relations

• Which of these relations are symmetric, reflexive
  or/and transitive?
• x lt y, x y (x, y are numbers)
• X Y, X and Y are disjoint (X,Y are sets)
• A is married to B (A, B are people)
• p is orthogonal to q (p, q are straight lines)
• P ? Q, P -gt Q (p, q are propositions)
• Someone can travel from x to y by walking and
  taking lift but not leaving a building
• (x, y are rooms)
• X and Y star in the same film (X, Y are
  actors/actresses)
• X is the ancestors of Y (X, Y are people)
• x and y do not have common factors (x, y are
  integers)

• Are the following relations equivalence
  relations? If yes, what are the equivalence
  classes?
• x and y have the same age/sex
• (x and y are people)
• There are lectures of x and y on the same day of
  the week (x, y are courses)
• x and y are partners in the same project group
  (x, y are students taking CSC2110)
• x and y are married (x, y are people)
• x and y are sibling (brother or sister) of each
  other (x, y are people)

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Tips and feedback

• For each concept mentioned in this tutorial, try
  to find your own examples
• The diagram representations (the Venn diagram,
  the diagrams of functions/relations) are usually
  more concrete and easy to understand
• We will discuss the classwork next week
• Take a look at it first and ask me next time if
  you have questions
• Some topics like the club strangers problem,
  the halting problem, uncountability of real
  numbers may be a bit more difficult, let me know
  if you want more explanation