Set Theory

Set
Theory
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What is a set?
 Sets are used to define the concepts of relations and functions.
The study of geometry, sequences, probability, etc. requires the
knowledge of sets.
 The theory of sets was developed by German
mathematician Georg Cantor (1845-1918).
 He first encountered sets while working on
“problems on trigonometric series”.
 Studying sets helps us categorize information.
It allows us to make sense of a large amount
of information by breaking it down into
smaller groups.
Georg Cantor
(1845-1918)

Definition: A set is any collection of objects specified in such
a way that we can determine whether a given object is or is
not in the collection.
 In other words A set is a collection of objects.
 These objects are called elements or members of the set.
 The symbol for element is .
 For example, if you define the set as all the fruit found in
my refrigerator, then apple and orange would be elements
or members of that set.
 The following points are noted while writing a set.
 Sets are usually denoted by capital letters A, B, S, etc.
 The elements of a set are usually denoted by small letters a,
b, t, u, etc
Examples:
 A = {a, b, d, 2, 4}
 B = {math, religion, literature, computer science}
 C = { }

Sets
 Sets can be well defined.
 A well defined set is a set whose contents are clearly
determined. The set defined as “colors” would not be
well defined while “the set of colors in a standard box of
eight crayons” is well defined.
 Other ways to denote sets
 Ellipses
 N = {0, 1, 2, 3, 4. . .}
(set of natural numbers)
 Z = {. . ., -3, -2, -1, 0, 1, 2, 3,. . .}
(set of integers)
 E = {0, 2, 4, 6. . .}
(set of even natural numbers)

There are three methods used to indicate a set:
1. Description
2. Roster form
3. Set-builder notation
 Description : Description means just that, words
describing what is included in a set.
 For example, Set M is the set of months that start with the
letter J.
 Roster Form : Roster form lists all of the elements in the
set within braces {element 1, element 2, …}.
 For example, Set M = { January, June, July}
 Set-Builder Notation: Set-builder notation is frequently
used in algebra.
 For example, M = { x x is a month of the year and x starts
with the letter J}
 This is read, “Set M is the set of all the elements x such that
x is a month of the year and x starts with the letter J”.

Subsets
 A is a subset of B if every element of A is also contained
in B. This is written
A B.
For example, the set of integers
{ …-3, -2, -1, 0, 1, 2, 3, …}
is a subset of the set of real numbers.
Formal Definition:
A B means “if x A, then x B.”
Empty set
 Set with no elements
 { } or Ø.
Elements may be sets
A = {1,2,{1,3,5},3,{4,6,8}}
B = {{1,2,3},{4,5,6}}
C = {Ø, 1, 3} = {{},1,3}
D = {Ø} = {{}} Ø

 Set size
 Called cardinality
 Number of elements in set
 Denoted |A| is the size of set A
 If A = {2,3,5,7,8}, then |A| = 5
 If a set A has a finite number of elements, it is a finite
set.
 A set which is not finite is infinite.
Set relations
 - "is a member of"
 x A
 - "subset"
 A B A is a subset of B
 Every element in A is
also in B
 x: x A x B

 - "proper subset"
 A B - A is a proper subset of B (A B)
 Every element in A is also in B and
 A B
 ( x: x A x B) A B
 - "proper superset"
 A B - A is a proper superset of B (A B)
 Every element in B is also in A and
 A B
 ( x: x B x A) A B
 Example: N Z Q R
 - "superset"
 A B
 A is a superset of B
 Every element in B is also in A
 x: x B x A

Numbers and Set
 There are different types of numbers:
 Cardinal numbers - answer the question “How many?”
 Ordinal numbers - such as first, second, third. . .
 Nominal numbers – which are used to name things.
Examples of nominal numbers would be your driver’s
license number or your student ID number.
 The cardinal number of a set S, symbolized as n(S), is the
number of elements in set S.
 If S = { blue, red, green, yellow } then n(S) = 4.
 Two sets are considered equal sets if they contain exactly
the same elements.
 Two sets are considered equivalent sets if they contain the
same number of elements ( if n(A) = n(B) ).

 If E = { 1 , 2 , 3 } and F = { 3 , 2 , 1 }, then the sets are
equal (since they have the same elements), and equivalent
(since they both have 3 elements).
 If G = { cat , dog , horse , fish } and H = { 2 , 5 , 7 , 9
}, then the sets are not equal (since they do not have the
same elements), but they are equivalent (since they both
have 4 elements, n(G) = n(H) ).
Power Sets
 Given any set, we can form a set of all possible subsets.
 This set is called the power set.
 Notation: power set or set A denoted as P(A)
 Ex: Let A = {a}
 P(A) = {Ø, {a}}
 Let A = {a, b}
 P(A) = {Ø, {a}, {b}, { a, b}}
• Let B = {1, 2, 3}
P(B)={Ø,{1},{2},{3},{1,2},{1,3}
,{2,3},{1,2,3}}

Cartesain Product
 Ordered pairs - A list of elements in which the order is
significant.
 Order is not significant for sets!
 Notation: use round brackets.
• (a, b)
{a,b} = {b,a}
(a,b) (b,a)
 Cartesian Product: Given two sets A and B, the set of
– all ordered pairs of the form (a , b) where a is any
– element of A and b any element of B, is called the
– Cartesian product of A and B.
 Denoted as A x B
• A x B = {(a,b) | a A and b B}
• Ex: Let A = {1,2,3}; B = {x,y}
– A x B = {(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)}
– B x A = {(x,1),(y,1),(x,2),(y,2),(x,3),(y,3)}
– B x B = B2 = {(x,x),(x,y),(y,x),(y,y)}
• (1, 2) • (2, 1)

Set Operators
 Union of two sets A and B is the set of all elements in
either set A or B.
 Written A B.
 A B = {x | x A or x B}
 Intersection of two sets A and B is the set of all elements
in both sets A or B.
 Written A B.
 A B = {x | x A and x B}
 Difference of two sets A and B is the set of all elements
in set A which are not in set B.
– Written A - B.
– A - B = {x | x A and x B}
– also called relative complement

 Complement of a set is the set of all elements not in
the set.
– Written Ac
– Need a universe of elements to draw from.
– Set U is usually called the universal set.
– Ac = {x | x U - A }
 Sets with no common elements are called disjoint
– If A B = Ø, then A and B are disjoint.
 If A1, A2, . . . An are sets, and no two have a common
element, then we say they are mutually disjoint.
– Ai Aj = Ø for all i,j n and i j
– Consider Md = {x | x MVNC students, d MVNC dorm
rooms}
– Consider Mn = {x I | (x MOD 5) = n}

 Partition - A collection of disjoint sets which
collectively
– Make up a larger set.
– Ex: Let A = {a,b}; B = {c,d,e}; C = {f,g} and
 D = {a,b,c,d,e,f,g}
 Then sets A,B,C form a partition of set D
 Let A be a nonempty set (A Ø), and suppose that
B1, B2, B3, . . ., Bn are subsets of A, such that:
– None of sets B1, B2, B3, . . ., Bn are empty;
– The sets B1, B2, B3, . . ., Bn are mutually disjoint.
(They have no elements in common)
– The union of sets B1, B2, B3, . . ., Bn is equal to A.
e.g. B1 B2 B3 . . . Bn = A

 Then we say the sets B1, B2, B3, . . ., Bn form a partition of
the set A.
 The subsets B1, B2, B3, . . ., Bn are called blocks of the
partition.
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1
7
2
3
8
9
4
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Universal Set
 A universal set is the super set of all sets under
consideration and is denoted by U.
 Example: If we consider the sets A, B and C as the
cricketers of India, Australia and England respectively,
then we can say that the universal set (U) of these sets
contains all the cricketers of the world.
 The union of two sets A and B is the set which contains
all those elements which
 are only in A, only in B and in both A and B, and this set
is denoted by “A B”.
 A B {x : x A or x B}
 Example: If A = {a, 1, x, p} and B = {p, q, 2, x},
then A B = {a, p, q, x, 1, 2}.
Here, a and 1 are contained only in A; q and 2 are
contained only in B; and p and x are contained in both A and B.

Set Properties
 Property 1 (Properties of Ø and )
– A Ø = A , A U = A
– A U = U , A Ø = Ø
 Property 2 ( The idempotent properties)
– A
 Property 3 (The commutative properties)
– A
 Property 4 (The associative properties)
– A C) = (A C
– A C) = (A B) C
 Property 5 (The distributive properties)
– A C) = (A B) C)
– A C) = (A B) (A C)

 Property 6 (Properties of the complement)
– ØC = U , UC = Ø
– A C = U , A AC = Ø
– (AC)C = A
 Property 7 (De Morgan's laws)
– (A B)C = AC BC
– (A B)C = AC BC
 Property 8 (Absortion laws)
– A (A B) = A
– A (A B) = A

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