MTH102 | Mth102 Short Lecture 01. What is Set? How many types of set?

MTH102 | Mth102 Short Lecture 01. What is Set? How many types of set?

Sets and Numbering Systems
The study of Mathematics begins with a study of sets and the development of the numbering
systems. Every mathematical system can be represented as a “set”; therefore, it is important for us
to understand the definitions, notations and properties of “sets”
• Definition: A set is an unordered collection of distinct objects. Objects in the collection are called
elements of the set.
• Examples:
o The collection of persons living in Lahore is a set.
 Each person living in Lahore is an element of the set.
o The collection of all towns in the Punjab province is a set.
 Each town in Punjab is an element of the set.
o The collection of all quadrupeds is a set.
 Each quadruped is an element of the set.
o The collection of all four-legged dogs is a set.
 Each four-legged dog is an element of the set.
o The collection of counting numbers is a set.
 Each counting number is an element of the set.
o The collection of pencils in your bag is a set.
 Each pencil in your bag is an element of the set.
• Notation: Sets are usually designated with capital letters. Elements of a set are usually designated
with lower case letters.
o D is the set of all four legged dogs.
o An individual dog might then be designated by d.
• The roster method of specifying a set consists of surrounding the collection of elements with
braces. For example the set of counting numbers from 1 to 5 would be written as
{1, 2, 3, 4, 5}.
• Set builder notation has the general form {variable | descriptive statement }.
The vertical bar (in set builder notation) is always read as “such that”.
Set builder notation is frequently used when the roster method is either inappropriate or
inadequate.
For example, {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6.
Note this is the same set as {1,2,3,4,5}.
• Other Notation: If x is an element of the set A, we write this as x ∈ A. x ∉ A means x is not an
element of A.
If A = {3, 17, 2 } then 3 ∈ A, 17 ∈ A, 2 ∈ A and 5 ∉ A.
If A = { x | x is a prime number } then 5 ∈ A, and 6 ∉ A.
• Definition: The set with no elements is called the empty set or the null set and is designated with
the symbol ∅.
• Definition: The universal set is the set of all things pertinent to a given discussion and is
designated by the symbol U
For example, when dealing with all the students enrolled at the Virtual University, the Universal
set would be
U = {all students at the Virtual University}
Some sets living in this universal set are:
A = {all Computer Technology students}
B = {first year students}
C = {second year students}
• Definition: The set A is a subset of the set B, denoted A ⊆ B, if every element of A is an element
of B.
If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B.
It is denoted by A ⊂ B.
If A is not a subset of B we write A ⊄ B to designate that relationship.
• Definition: Two sets A and B are equal if A ⊆ B and B ⊆ A. If two sets A and B are
equal we write A = B to designate that relationship.
In other words, two sets, A and B, are equal if they contain the same elements
• Definition: The intersection of two sets A and B is the set containing those elements which are
elements of A and elements of B. We write A ∩ B to denote A Intersection B.
• Example: If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A ∩ B = {3, 6}
• Definition: The union of two sets A and B is the set containing those elements which are
elements of A or elements of B. We write A ∪ B to denote A Union B.
• Example: If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6}.