UCCM 1353
Basic Algebra
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Chapter 3
Set Theory
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Set
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♣ Well-defined list, collection, or class of objects.
♣ Objects in a set are called elements or
members.
♣ Describe:
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List all (if possible)
State those properties which characterize the
elements, A = {x | x is even number}
Venn Diagram
Set (Cont…)
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╔ Empty Sets
╠ Equality Sets
╠ Finite and Infinite Sets
╠ Subsets
╠ Power Set
╚ Family of sets
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Empty Set
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Also called Null Set
Contain no elements
Denoted by Ø
E.g: {x | x2 = 4, x is odd}
{x R | x2 = −1}
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Finite and Infinite Sets
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͏ exactly n distinct elements in set S where n
is a nonnegative integer, then S is a finite
set and n is the cardinality of S.
͏ The cardinality of S is denoted by |S|.
͏ E.g:
S = {a, b, c, 1, 1, 3, 4}
|S| = 6 and S is a finite set.
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Finite and Infinite Sets
(Cont…)
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Examples
Identify whether the following sets are finite
or infinite.
The set of odd numbers
The set of months of the year
S = { x R+ | −5 < x < 5}
S = {x Z | −5 < x < 5 }
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Equality Sets
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Set A and B are equal if they have same
members. A = B
E.g:
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a) {a, b, e, h, k} = {a, b, c, h, k} ?
b) {1, 2, 3} = {3, 2, 1} ?
c) {1, 2, 3} = {3, 2, 1, 1} ?
d) {x | x2 + 5x = −6} = {2, 3} ?
e) A = Set of all prime numbers
B = Set of all odd numbers
A=B?
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Equality Sets (cont…)
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E.g:
A = {n Z | n = 2p for some integer p}
B = the set of all even integers
C = {m Z | m = 2q – 2 for some integer q}
D = {k Z | k = 3r + 1 for some integer r}
A = B? A = C? A = D?
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A = {…, −4, −2, 0, 2, 4,…}
B = {…, −4, −2, 0, 2, 4,…}
C = {…, −4, −2, 0, 2, 4,…}
D = {…, −5, −2, 1, 4, 7,…}
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Subsets
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A B, A subset of B, iff every element of A
is also an element of B.
For any set A, Ø A and A A
If A B and B C, then A C
E.g:
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A = { x | x is a positive power of 2}
B = { x | x is even}
C = { x | x is a multiple of 4}
A B?
B C?
AC?
Subsets (Cont…)
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#
elements
Set
Subsets
#
subsets
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{}
{}
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1
{a}
{ }, {a}
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{a, b}
{ }, {a}, {b}, {a, b}
3
{a, b, c}
{ }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c},
{a, b, c}
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{a, b, c, d}
……….
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Subsets (Cont…)
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A B, A proper subset of B, iff every element
of A is in B (A B) but there is at least one
element of B that is not in A (A ≠ B).
E.g:
A = {x N | x is a prime # and x < 13}
B = {2, 3, 5, 7}
a) Is A B? If so is A B?
b) Is B A? If so is B A ?
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Subsets (Cont…)
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E.g:
Let A = {B76, XR3, D54, ES2},
B = {B76, D54} and
C = {ES2, XL5}.
B A?
C A?
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B B?
Subsets (Cont…)
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Difference between subset and proper subset:
A subset can be the set itself, like A is a subset of A. A
proper subset means that the subset is not the set itself.
So A is not a proper subset of A.
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let A ={1, 2, 3}, B={1, 2, 3}, C={2, 3}
B even though is the same as A, it is also a subset of A.
(Every element of B is also an element of A.)
C is a proper subset of A. (Every element of C is also an
element of A, but C is not equal to A. There is no 1 in C.)
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Venn Diagram
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Universal set, U
Set A
Elements
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Set B
Proper
Subset
Elements
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Sets of sets (Family of sets)
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• Elements of a set are sets themselves.
• E.g:
S = {{2, 5, 6}, {1, 3}, {6, 9}}
A = {{Physic, History}, {Math, English}}
B = {{x | x is a multiple of 2}, {x | x is a
multiple of 8}}
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Power Set
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Power set of S, P(S) is the set of all subsets of
the set S.
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Set
Subsets
Power Set P(S)
{}
{}
{Ø}
{a}
{ }, {a}
{Ø, {a}}
{a, b}
{ }, {a}, {b}, {a, b}
{Ø, {a}, {b}, {a, b}}
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{ }, {a}, {b}, {c}, {a, b}, {Ø, {a}, {b}, {c}, {a, b},
{a, b, c}
{a, c}, {b, c}, {a, b, c}
{a, c}, {b, c}, {a, b, c}}
Questions
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Let A = {1, 3}, B = {3, 1}, C = {1, 3, 4}
State True / False
A = B True
A ⊂ B False
C ⊆ A False
A ⊆ B True
A ⊆ C True
C ⊂ C False
B ⊆ A True
A ⊂ C True
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Questions
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1. {1, 2, 3, 4} {x N | x < 5} ?
2. {1, 2, 3, 4} {x N | x < 5} ?
3. {Ø} { } ?
4. {} {Ø} ?
5. {5} {1, 2, 3, 4, 5} ?
6. 6 {3, 6, 9, 12} ?
7. {1, 2, 3, 4, 5} = {a, b, c, d, e} ?
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Cartesian Products
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A × B, Cartesian product of A and B, is the
set of all ordered pairs (a, b) where a A
and b B.
E.g:
A = {1, 2}, B = {a, b}
A × B = {(1, a), (1, b), (2, a), (2, b)}
B × A = {(a, 1), (a, 2), (b, 1), (b, 2)}
A×B≠B×A
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Operations
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Union
Intersection
Difference
Complement
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Union
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A B, A union B, is the set of all elements
which belong to A or to B or to both.
E.g:
A = {a, b, c, d}, B = {e, f, g, h}
A
A B = {a, b, c, d, e, f, g, h}
a
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Note: A B = B A
A A B and B A B
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c db
e f
g h
B
Intersection
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A B, A intersect B, is the set of elements
which are common to A and B.
E.g:
A = {a, b, c, d}, B = {e, b, d, f}
A
A B = {b, d}
a
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c
Note: A B = B A
(A B) A and (A B) B
A B = Ø if A and B are disjoint
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e
f
B
Difference
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A – B, difference of sets A and B, is the set
of elements which belong to A but do not
belong to B.
E.g:
A = {3, 5, 8, 10}
B = {1, 5, 7, 10}
A
A – B = {3, 8}
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B
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B – A = {1, 7}
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Difference (Cont…)
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Note:
i. (A – B) A, (A – B) B = Ø
ii. (A – B) (A B) = (A – B) (B − A) =
(A B) (B – A) = Ø
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A
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B
Complement
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A∁, complement of set A, is the set of
elements which do not belong to A.
E.g:
Universal set ={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 5, 6, 8, 9}
A∁ ={1, 3, 4, 7, 10}
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Complement (Cont…)
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Note:
i. A A∁ = U , A A∁ = Ø
ii. U∁ = Ø , Ø∁ = U
iii. (A∁)∁ = A
A
iv. A – B = A B∁
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U
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B
Extra
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Binary operation
Union () and intersection () which
combine two sets from within the same
universal set U to form a third set in U.
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Unary operation
Operation on a single set A, such as forming
complement A, A∁
Comparable sets
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A and B are called comparable if A B or
B A.
U
B
‡ A B implies A B = A.
A
‡ A B implies A B = B.
‡ A B implies Bc Ac.
‡ A B implies A (B – A) = B.
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U
A
B
B
Set Identities
U
A
C
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Associative Laws
(A B) C = A (B C)
(A B) C = A (B C)
Distributive Laws
A(BC) = (AB)(AC)
A(BC) = (AB)(AC)
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(A B)∁ = A∁ B∁
Absorption Laws
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De Morgan’s Laws (A B)∁ = A∁ B∁
A (A B) = A
A (A B) = A
Set Identities (Cont…)
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Idempotent Laws
AA=A
AA=A
Domination Laws
AØ=Ø
AU=U
Identity Laws
AU=A
AØ=A
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Set Identities (Cont…)
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Complementation Law
A A∁ = U
Exclusion Law
A A∁ = Ø
Double Complementation Law
Alternate representation
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(A∁)∁= A
A − B = A B∁
Set Identities (Cont…)
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E.g:
Prove (A B) – C = (A – C) (B – C)
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(A B) – C = (A B) C∁ Alternate rep.
= C∁ (A B)
Associative
= (C∁ A) (C∁ B) Distributive
= (A C∁) (B C∁)
= (A − C) (B − C)
Alternate rep.
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Set Identities (Cont…)
E.g: Prove (A B)c = Ac Bc
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Suppose x (A B)c
So x A B
x A and x B
x Ac and x Bc
x Ac Bc
(A B)c Ac Bc
Suppose
x Ac Bc
x Ac and x Bc
x A and x B
xAB
x (A B)c
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Ac Bc (A B)c
(A B)c Ac Bc and Ac Bc (A B)c
Ac Bc = (A B)c
Partitions of Sets
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Nonempty sets {A1, A2, …, An} is a partition of a set A iff
(i) A = A1 A2 … An
(ii) A1, A2, …, An are mutually disjoint.
Disjoint
A
A1
A3
A2
A4
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Two sets are called disjoint iff they have no elements in common.
Given A = {1, 3, 5} and B = {2, 4, 6}, A and B are disjoint.
Mutually Disjoint
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The sets A1, A2, … , An are mutually disjoint iff no two sets Ai and Aj
with distinct subscripts have any elements in common.
Given X = {2, 4, 6}, Y = {3, 7} and Z = {4, 5}.
X, Y and Z are not mutually disjoint since X and Z are not disjoint.
(even though X and Y disjoint, Y and Z disjoint)
Russell’s Paradox
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Most sets are not elements of themselves.
Some sets are elements of themselves.
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The container (Set) that contains all (containers)
that don't contain themselves.
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If R is a member of itself, then by definition it
must not be a member of itself. Similarly, if R is
not a member of itself, then by definition it must
be a member of itself.
Russell’s Paradox (Cont…)
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Giving a person a piece of paper on which
is written: ‘The statement on the other side
of this paper is false.’ The person turns the
paper over, and finds on the other side: ‘The
statement on the other side of this paper is
true.’
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Halting Problem
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Given a description of a program, decide whether
the program finishes running or continues to run,
and will thereby run forever. This is equivalent to
the problem of deciding, given a program and an
input, whether the program will eventually halt
when run with that input, or will run forever.
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E.g: 1/3, 1/6
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Halting Problem (Cont…)
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Potential solutions to halting problem
Run program for a really long time and see if it
stops
Analyze code to see if there are infinite loops
Check if loop exit conditions become closer to
being met over time
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Halting Problem (Cont…)
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Halting behaviors
Program halts
Program loops but keeps repeating itself
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Sometimes detectable, unless loop is really long
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Program continues changing without repetition
Hard to tell if it will stop or start repeating or keep
changing forever
Computer Representation of Sets
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Method for storing elements using an arbitrary
ordering of the elements of the universal set.
Specify an arbitrary ordering of the elements
of U, for instance a1, a2, . . . , an.
Represent a subset A of U with the bit string of
length n, where the ith bit in this string is 1
if ai belongs to A and is 0 if ai does not belong
to A.
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Computer Representation of Sets
(Cont…)
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E.g:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Bit strings represent the subset of odd integers in U:
{1, 3, 5, 7, 9}, has a one bit in the first, third, fifth, seventh,
and ninth positions. It is 10 1010 1010.
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Bit string that represent the subset of even integers in U :
{2, 4, 6, 8, 10}. It is 01 0101 0101.
All integers in U that do not exceed 5, {1, 2, 3, 4, 5}, is
represented by the string 11 1110 0000.
Computer Representation of Sets
(Cont…)
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E.g:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Complement of {1, 3, 5, 7, 9}, replacing 0s with 1s for
10 1010 1010, yields 01 0101 0101 which corresponds
to the set {2, 4, 6, 8, 10}.
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The bit strings for the sets {1, 2, 3, 4, 5} and
{1, 3, 5, 7, 9} are 11 1110 0000 and 10 1010 1010.
The bit string for the union of these sets is 11 1110
0000 ∨ 10 1010 1010 = 11 1110 1010, which
corresponds to the set {1, 2, 3, 4, 5, 7, 9}.
Computer Representation of Sets
(Cont…)
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E.g:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
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The bit strings for the sets {1, 2, 3, 4, 5} and
{1, 3, 5, 7, 9} are 11 1110 0000 and 10 1010 1010.
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The bit string for the intersection of these sets is
1111100000 ∧ 1010101010 = 10 1010 0000,
which corresponds to the set {1, 3, 5}.
Countably Infinite
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A countable set is a set with the same cardinality
(number of elements) as some subset of the set of
natural numbers.
A countable set is either a finite set or a countably
infinite set. Whether finite or infinite, the elements of a
countable set can always be counted one at a time and,
although the counting may never finish, every element of
the set is associated with a natural number.
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E.g: {0, 1, −1, 2, −2, 3, −3, …}
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Countably Infinite
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Countably infinite is in contrast to uncountable, which
describes a set that is so large, it cannot be counted even if
we kept counting forever.
A set is uncountable if it contains so many elements that
they cannot be put in one-to-one correspondence with the
set of natural numbers. In other words, there is no way that
one can count off all elements in the set in such a way that,
even though the counting will take forever, you will get to
any particular element in a finite amount of time.
E.g: The set of real numbers in the interval [0,1] is
uncountable
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End
of
Chapter 3
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