1.2.1 Set Notation
Set:
A set is a collection of well-defined and distinct objects. It can be described by putting its components
e.g.
A={1,2,3,4,5,6}
Elements:
The elements of a set are the items in it. Elements can be anything – numbers, characters, locations, and so on.
To represent is an element of we use the symbol \( \in \)
e.g.
If \( N= \lbrace{1,2,3,4}\rbrace \) then \( 1,2,3,4\in N \)
Universal Set:
The universal set is a set that contains all of its subsets’ elements, as well as its own.
It is represented by capital letter U or by Greek letter \( \xi \) (xi)
Empty Set:
The set that does not contain any element is called empty set.
e.g.
A set of natural numbers less than \( 1 \).
It is expressed as \( \lbrace{}\rbrace \) and represented by Greek letter \( \phi \)
Number of elements:
Number of elements in a set \( A \) is denoted by \( n(A) \)
e.g.
If \(A= \lbrace{a,e,i,o,u}\rbrace \) then \( n (A)=5 \)
Note that
If \( B= \phi \) then \( n (B)=0 \)
If \( C= \lbrace{0}\rbrace \) then \( n (C)=1 \)
Subset:
If every element of set \( A \) is in set \( B \) then set \( A \) is a subset of set \( B \).
It is further divided into two parts
Proper subset
If all elements of \( B \) are in \( A \) but \( A \) contains at least one element that is not in \( B \) then \( A \) is a proper subset of \( B \). It is denoted as \( A \subset B \)
e.g.
\( A= \lbrace{blue, green, purple, pink }\rbrace \)
\( B= \lbrace{pink, purple, green, maroon, blue}\rbrace \)
So \( A\subset B \)
Improper subset
Any subset \( A \) which contains all the elements of the original set \( B \) is called an improper subset. It is represented as \( A \subseteq B \)
e.g.
\( X= \lbrace{a,b,c,d,e}\rbrace \), \( Y= \lbrace{b,d,e,a,c}\rbrace \)
So \( X \subseteq Y \)
A cross through the sign indicates that it is not true.
\( \neq \) is not equal to
∉ does not belong to
\( ⊈ \)neither a subset of nor equal to
\( ⊄ \)not a subset of
Set operations:
There are three major types of set operations.
Union of sets
Union of two sets \( A \) and \( B\) is a set which contains all the elements of set \( A \) and set \( B \). And it is
represented as \( A \) 
\( B \)
e.g.
If \(A= \lbrace{1,3,5,7,9,11,…}\rbrace \) and \( B= \lbrace{2,4,6,8,10,…}\rbrace \)
then
\( A \) \( B = \lbrace{1,2,3,4,5,6,7,8,9,10,11, …..}\rbrace \)
Intersection of sets
Intersection of two sets \( A \) and \( B \) is a set which contains all the common elements of set \( A \) and set \( B \). And it is represented as \( A \)∩\( B \).
e.g.
If \( A= \lbrace{2,3,5,7,11,13,…}\rbrace \) and \( B= \lbrace{2,4,6,8,10,…}\rbrace \)
then
\( A \)∩\( B \)\( = \lbrace{2}\rbrace \)
Complement of a set
If \( U \) is a universal set and \( A \) is any subset of \( U \) then the complement of \( A \) is the set of all elements of the set \( U \) apart from the elements of \( A \). It is represented as \( A’ \) or \( A^c \)
e.g.
If \( U= \lbrace{1,2,3,4,5,6,7,8,9,10,…}\rbrace \) and \( A= \lbrace{2,4,6,8,10,…}\rbrace \)
then
\( A’= \lbrace{1,3,5,7,9,11,…}\rbrace \)
Properties of set operation:
Commutative property
\( A \)∩\( B \)=\( B \)∩\( A \)
\( A \)\( B \)=\( B \)\( A \)
Associative property
\( A∩ (B∩C) = (A∩B)∩C \)
\( A \) (\( B \) \( C \) )=(\( A \) \( B \)) \( C \)
Distributive property
\( A \) \( (B ∩ C) \)=(\( A \) \( B \)) ∩ (\( A \) \( C \))
\( A \)∩ (\( B \) \( C \))= (\( A∩ B \)) (\( A∩ C \))