ITP3902_DMS_Lec_1_Set_Operations 050915 
Lecture 1 Set Operations & Applications
Lecture 1
Set Operations & Applications
ITP3902 Discrete Mathematics & Statistics
Page 1
Lecture 1 Set Operations & Applications
Sets and elements
•
A Set is a collection of objects. Often the objects in a set have similar properties.
• Set theory is the foundation of modern mathematics.
Element
• The objects in a set are called its elements.
• If a is an element of set A, it belongs to A and is written as a
A.
• If a does not belong to A, we write a A.
ITP3902 Discrete Mathematics & Statistics
Page 2
Lecture 1 Set Operations & Applications
Sets and elements (Cont’d)
• We usually use
• uppercase letters A, B, C, ......, to denote sets, and
• lowercase letters a, b, c, ......, to denote the elements of a set. • Two ways to specify a set
• by listing all the elements that belong to the set; or
• by specifying the characteristic properties of the elements in the set in a statement.
ITP3902 Discrete Mathematics & Statistics
Page 3
Lecture 1 Set Operations & Applications
Sets and elements (Cont’d)
Example 1
A = { a, e, i, o, u } is a set and is specified by listing all its elements.
Example 2
B = { x : x is an integer, x > 0 } is a set specified by a property statement. We may also list its elements as B = { 1, 2, 3, 4, … }.
ITP3902 Discrete Mathematics & Statistics
Page 4
Lecture 1 Set Operations & Applications
Equality of sets
• Two sets A and B are equal if and only if they have exactly the same elements i.e. if every element belonging to A also belongs to B and every element belonging to B also belongs to A.
• Note
A set does not have duplicate elements.
Example 3
Let A = { a, e, i, o, u }, B = { u, o, i, e, a } and
C = { a, a, e, i, i, o, u } then A = B = C.
Note that the elements in C are not properly listed. The example just serves to illustrate that a set does not have duplicate elements.
ITP3902 Discrete Mathematics & Statistics
Page 5
Lecture 1 Set Operations & Applications
Universal sets
•
Lecture 1
Set Operations & Applications
ITP3902 Discrete Mathematics & Statistics
Page 1
Lecture 1 Set Operations & Applications
Sets and elements
•
A Set is a collection of objects. Often the objects in a set have similar properties.
• Set theory is the foundation of modern mathematics.
Element
• The objects in a set are called its elements.
• If a is an element of set A, it belongs to A and is written as a
A.
• If a does not belong to A, we write a A.
ITP3902 Discrete Mathematics & Statistics
Page 2
Lecture 1 Set Operations & Applications
Sets and elements (Cont’d)
• We usually use
• uppercase letters A, B, C, ......, to denote sets, and
• lowercase letters a, b, c, ......, to denote the elements of a set. • Two ways to specify a set
• by listing all the elements that belong to the set; or
• by specifying the characteristic properties of the elements in the set in a statement.
ITP3902 Discrete Mathematics & Statistics
Page 3
Lecture 1 Set Operations & Applications
Sets and elements (Cont’d)
Example 1
A = { a, e, i, o, u } is a set and is specified by listing all its elements.
Example 2
B = { x : x is an integer, x > 0 } is a set specified by a property statement. We may also list its elements as B = { 1, 2, 3, 4, … }.
ITP3902 Discrete Mathematics & Statistics
Page 4
Lecture 1 Set Operations & Applications
Equality of sets
• Two sets A and B are equal if and only if they have exactly the same elements i.e. if every element belonging to A also belongs to B and every element belonging to B also belongs to A.
• Note
A set does not have duplicate elements.
Example 3
Let A = { a, e, i, o, u }, B = { u, o, i, e, a } and
C = { a, a, e, i, i, o, u } then A = B = C.
Note that the elements in C are not properly listed. The example just serves to illustrate that a set does not have duplicate elements.
ITP3902 Discrete Mathematics & Statistics
Page 5
Lecture 1 Set Operations & Applications
Universal sets
•