


Definition 
A group of objects which have the common characteristics and classified in the same group.



Describe sets: 

 Description
 Listing
 Set builder notation


Example 
Describe the multiples of \(3\) which are less than \(19\) by using;
(i) description
(ii) listing
(iii) set builder notation

(i)
Let the set be represented by \(R\).
Description: \(R\) is the set of multiples of \(3\) which are less than \(19\).

(ii)
Listing: \(R=\{3, 6, 9, 12, 15, 18\}\)

(iii)
Set builder notation:
\(R=\{x:x\text{ is the multiple of 3 and }x\lt19\}\)



Empty set: 

Definition 
A set that contains no elements.



 An empty set can be represented with the symbol \(\phi\) or { }.
 An empty set is also called a null set.
 The symbol \(\phi\) is read as phi.


The element of a set: 

Definition 
Each object in a set.



 Each of the elements must satisfy the conditions of the set that is defined.
 Symbol \(\in\) (epsilon) is used to represent ‘is an element of’ the set.
 Symbol \(\notin\) is used to represent ‘is not an element of’ the set.


Determine the number of elements of a set: 

 Number of elements in set \(P\) can be represented by the notation \(n(P)\).
 List all the elements in a set so that the number of elements in the set can be determined.


Example 
Determine the number of elements in the following set.
\(A=\{\text{colours of the traffic light\}}\)

Noted that
\(\begin{aligned}A&=\{\text{colours of the traffic light\}} \\\\&=\{\text{red, yellow, green\}}. \end{aligned}\)
Thus, \(n(A)=3\).



Equality of sets: 

Definition 
Sets in which every element of the sets are the same.



 If every element in two or more sets are the same, then all the sets are equal.
 The order of elements in a set is not important.
 Symbol \(\neq\) means ‘is not equal to’.


Example 
Determine whether the following pair of sets is an equal set.
\(\begin{aligned}S&=\{\text{letters in the word 'AMAN'}\} \\\\T&=\{\text{letters in the word 'MANA'}\} \end{aligned}\)

We can see that
\(\begin{aligned}S&=\{\text{A, M, A, N}\} \\\\T&=\{\text{M, A, N, A}\}. \end{aligned}\)
Each element in set \(S\) is equal to each element in set \(T\).
Thus, \(S=T\).


