SLIDE 1
Sets
Definition (Set)
A set is a collection of object.
Alan H. SteinUniversity of Connecticut
Notation
We may define a set by listing its elements between {curly braces}.
Alan H. SteinUniversity of Connecticut
Sets Definition (Set) A set is a collection of object. Alan H. - - PowerPoint PPT Presentation
Sets Definition (Set) A set is a collection of object. Alan H. - - PowerPoint PPT Presentation
Sets Definition (Set) A set is a collection of object. Alan H. SteinUniversity of Connecticut Notation We may define a set by listing its elements between { curly braces } . Alan H. SteinUniversity of Connecticut Notation We may define a set
SLIDE 2
SLIDE 3
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10}
Alan H. SteinUniversity of Connecticut
SLIDE 4
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10} We may define a set by listing the properties its elements must satisfy,
Alan H. SteinUniversity of Connecticut
SLIDE 5
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10} We may define a set by listing the properties its elements must satisfy, i.e. {x : p(x)},
Alan H. SteinUniversity of Connecticut
SLIDE 6
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10} We may define a set by listing the properties its elements must satisfy, i.e. {x : p(x)}, where p(x) describes the properties an element x must satisfy.
Alan H. SteinUniversity of Connecticut
SLIDE 7
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10} We may define a set by listing the properties its elements must satisfy, i.e. {x : p(x)}, where p(x) describes the properties an element x must satisfy. Example: {x : 2 ≤ x ≤ 10 and x is even.}
Alan H. SteinUniversity of Connecticut
SLIDE 8
Notation
We may define a set by listing its elements between {curly braces}. Example: {2,4,6,8,10} We may define a set by listing the properties its elements must satisfy, i.e. {x : p(x)}, where p(x) describes the properties an element x must satisfy. Example: {x : 2 ≤ x ≤ 10 and x is even.} Set Inclusion: x ∈ A means x is an element of the set A.
Alan H. SteinUniversity of Connecticut
SLIDE 9
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Alan H. SteinUniversity of Connecticut
SLIDE 10
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}.
Alan H. SteinUniversity of Connecticut
SLIDE 11
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Alan H. SteinUniversity of Connecticut
SLIDE 12
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Definition (Complement)
Ac = A′ = {x : x / ∈ A}.
Alan H. SteinUniversity of Connecticut
SLIDE 13
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Definition (Complement)
Ac = A′ = {x : x / ∈ A}. This must be understood in context.
Alan H. SteinUniversity of Connecticut
SLIDE 14
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Definition (Complement)
Ac = A′ = {x : x / ∈ A}. This must be understood in context. We always work within some universal set U.
Alan H. SteinUniversity of Connecticut
SLIDE 15
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Definition (Complement)
Ac = A′ = {x : x / ∈ A}. This must be understood in context. We always work within some universal set U. By Ac, we really mean the set of elements within U which are not in A.
Alan H. SteinUniversity of Connecticut
SLIDE 16
Operations on Sets
Definition (Union)
A ∪ B = {x : x ∈ A or x ∈ B}.
Definition (Intersection)
A ∩ B = {x : x ∈ A and x ∈ B}. Note that the words and and or have very different meanings.
Definition (Complement)
Ac = A′ = {x : x / ∈ A}. This must be understood in context. We always work within some universal set U. By Ac, we really mean the set of elements within U which are not in A.
Definition (Set Difference)
A − B = {x ∈ A : x / ∈ B}.
Alan H. SteinUniversity of Connecticut
SLIDE 17
DeMorgan’s Laws
Theorem (DeMorgan’s Laws)
(A ∪ B)c = Ac ∩ Bc, (A ∩ B)c = Ac ∪ Bc
Alan H. SteinUniversity of Connecticut