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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
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Set Theory

  • 2. 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)
  • 3. 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 = { }
  • 4. 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)
  • 5. 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”.
  • 6. 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
  • 7.  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
  • 8.  - "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
  • 9. 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) ).
  • 10.  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}}
  • 11. 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)
  • 12. 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
  • 13.  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}
  • 14.  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
  • 15.  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. 0 1 7 2 3 8 9 4 65
  • 16. 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.
  • 17. 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)
  • 18.  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
  • 19. The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit