01. Set Theory
01-1 Sets and Elements
(1) Set: A set is a group or collection of objects, things or symbols which can
be clearly defined.
(a) Examples: a group of students, a list of states in a country, a collection of baseball cards, etc.
(b) Capital letters are normally used to identify a set
(2) Elements: Individual objects in a set are called the members or elements
of the set.
A
a
b
If A  {a}, then
a  A and b  A
(3) Notations: The symbol “” is used to relate an element and a set.
(a)  denotes “is an element of”
(b)  denotes “is not an element of”
01-2 Describing Sets
(1) List or Roster Method: The set can be defined by listing all its elements, separated by
commas and enclosed within braces.
(2) Set-Builder Notation: The set can be defined by using a rule or semantic description.
(3) Venn Diagram: A Venn diagram is a visual diagram that shows the relationship of sets with
one another. In a Venn diagram, the sets are represented by shapes; usually circles or ovals.
[Example]
(1)
Roster method: A  {2, 4, 6, 8}
(2)
Set-builder notation: A  { x | 0  x  10, x is an even number}
(3) Venn diagram:
2
6
A
4
8
01-3 Finite and Infinite Sets
(1) Finite Sets: Finite sets are sets that have a finite number of elements.
ex.) A  {0, 2, 4, 6, …, 100}
(2) Infinite Sets: An infinite set is a set that is not finite. It is not possible to explicitly list out
all the elements of an infinite set.
ex.) A  { x | x is a natural number}
(3) Number of Elements
(a) The number of elements in a finite set A is denoted by n(A).
(b) Null or Empty Set: A set has no elements and is represented by the symbol { } or .  A    n(A)  0
ex.) n()  0, n({})  1, n({0})  1
(4) Universal Set: A universal set is the set of all elements under consideration, denoted by a
capital U.
(a) In a Venn diagram, the universal set is usually represented by a rectangle and labeled U.
[example]
(4)
 Given that U  {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
(i) A  { x | x is a factor of 60}  A  {5, 6, 10, 12}
(ii) B  { x | x is a prime number}  B  {5, 7, 11}
 Draw a Venn diagram to represent the following sets:
U  {1, 2, 3, 4, 5, 6, 7, 8, 9}, A  {1, 2, 5, 6}, B  {3, 9}
I. Number and Operations
01-4 Subsets
(1) Subsets: If every element of set A is also an element of set B,
then the set A is called a subset of B (A  B).
B
(a) Proper Subset: A is a proper subset of B if A is a subset of B, but B  A.
(b) Equal Set: A is an equal set of B if A  B and B  A
 A  B and B  A  A  B
(c) Power Sets: The power set of set A is the set of all subsets of set A
 P(A)  { X | X  A}
A
A  B, but B  A
 A is a proper subset of B
(2) Notations: The symbol “” means “is a subset of” and The symbol
“” means “is not a subset of”
ex.) A  {1, 3, 5}, B  {1, 2, 3, 4, 5}  Every element in A is also in B, so A  B
ex.) X  {1, 3, 5}, Y  {2, 3, 4, 5, 6}  1 is in X but not in Y, so X  Y
(3) Properties of Subsets
(a) Every set is a subset of itself  For any set A, A  A
(b) The empty set is a subset of any set A    A
(c) For any two sets A and B, if A  B and B  A then A  B
(d) For any three sets A, B, and C, if A  B and B  C, then A  C (Transitive Property)
(4) Number of Subsets: The number of subsets for a finite set A is given by 2 n(A)
[Note]
(3)
(b) Equal Subset: When two sets A and B contain exactly the same number of elements and the elements are the
same, A is equal to B and it is written as A  B.
ex.) If A  {1, 2, 3, 6} and B  {x | x is a factor of 6}, then A  B.
[Example]
(1)
(b) If A  {1, 2, 3, 6} and B  {x | x is a factor of 6}, then A  B.
(c) A  {1, 2}
 P(A)  { { }, {1}, {2}, {1, 2}}, P(A) has 4 elements.
(4)
 List all the subsets of set A  {1, 2, 3}
 The subsets of set A are { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}
The number of subsets is 2n(A)  23  8.
 List all the subsets that have an element of 1 among the subsets of set A  {1, 2, 3}  22  4
01-5 Set Operations – Union and Intersection
(1) Union of Sets: The union of two sets A and B, denoted by
A  B is the set of elements in set A or set B.
 A  B  { x | x  A or x  B}
(2) Intersection of Sets: The intersection of two sets A and B,
denoted by A  B is the set of elements in both set A and set B.
 A  B  { x | x  A and x  B}
(3) Properties of Unions and Intersections
For two sets A and B,
(a) A    , A    A
(b) A  A  A, A  A  A
(c) A  AC  , A  AC  U
(d) If A  B, A  B  A, A  B  B
(e) (A  B)  A, (A  B)  B
U
A
B
AB
U
A
B
AB
(f) A  (A  B), B  (A  B)
(g) Commutative property: A  B  B  A, A  B  B  A
A  (B  C)  (A  B)  C
A  (B  C)  (A  B)  C
(h) Distributive property: A  (B  C)  (A  B)  (A  C)
A  (B  C)  (A  B)  (A  C)
(4) Number of Elements in Union and Intersection
(a) Number of Elements in Union: n(A  B)  n(A)  n(B)  n(A  B)
(b) Number of Elements in Intersection: n(A  B)  n(A)  n(B)  n(A  B)
[Note]
 For three sets A, B, and C, n(A  B  C)  n(A)  n(B)  n(C) n(A  B)  n(B  C)  n(C  A)  n(A  B  C)
I. Number and Operations
01-6 Set Operations – Complement and Difference
(1) Complement of a Set: The complement of set A, denoted by AC, is
the set of all elements in the universal set that are not in A.
 AC  { x | x  U and x  A}
(2) Difference of A and B:
The difference of A and B, denoted by A  B, is the set of all elements
which are in A but not in B (relative complement of B in A).
 A  B  { x | x  A and x  B}  A  BC
(3) Properties of Complement of a Set
U
U
B
A
AC
A
B
AB
(a) UC  , C  U
(b) (AC)C  A
(4) Properties of Relative Complement
(a) A  B  B  A (open commutative property)
(b) AC  U  A
(c) A  B  A  BC  A  (A  B)  (A  B)  B
(d) If A  B   (Disjoint Sets), A  B  A
(e) A  B  A  B  A  A  B  B  A  B  A  BC  
(5) Number of Elements in Complement and Relative Complement
(a) n(A)  n(AC)  n(U)  n(AC)  n(U)  n(A)
(b) n(A  B)  n(A)  n(A  B)  n(A  B)  n(B)
(6) De Morgan’s Theorems
(a) (A  B)C  AC  BC
(b) (A  B)C  AC  BC
[Example]  How many integers in the set of all integers from 1 to 100, inclusive, are not the square of an integer? Ans.) 90
[Note]
 Defined Operators or Functions (Special Symbols Problems):
ex.) ^a is the sum of integers from 1 to a. What is ^^5?
 By definition of ^a  1  2  3  …  a, ^5  1  2  3  4  5  15
^^5  ^(^5)  ^15  1  2  3  …  15  120
ex.) a * b  1  (b  a), a & b  b  a
If (3 & 5) * (c & 8)  c & 10, then what is the value of c?
3&5532
c&88c
(3 & 5) * (c & 8)  1  (8  c  2)  1  (10  c)
(left side)  (right side): 1 / (10  c)  10  c, 1  (10  c)2, c2  20c  99  0, (c  9)(c  11)  0
 c  9 or 11
 Greatest Integer Function f(x)  [x]
For all real numbers x, the greatest integer function returns the largest
integer less than or equal to x. In other words, the greatest integer function
rounds down a real number to the nearest integer.
ex.) [5.9]  5, [7]  7, [8.9]  9
I. Number and Operations