
11.2 
Venn Diagrams, Universal Sets, Complement of a Set and Subsets 


Universal set: 

Definition 
A set that consists of all the elements under discussion.



 The symbol for universal set is \(\xi\).


Complementary set: 

Definition 
The elements in the universal set that are not the elements of the set.



Example 
Tthe following is the universal set and set \(P\).
\(\begin{aligned} \xi&=\{2, 3, 4, 5\} \\\\P&=\{2, 3, 5\}\end{aligned}\)
(i) State whether set \(P\) is a universal set.
(ii) Based on the universal set, determine the complement of set \(P\).

(i)
Set \(P\) is not a universal set as it does not contain element \(4\).

(ii)
The complement of set \(P\) is
\(P'=\{4\}\).



Represent the universal set and complement of a set by using Venn diagram: 

 A set can be represented by a circle, an oval, a rectangle or a triangle.
 The universal set is commonly represented by a rectangle.
 A set can also be represented by an enclosed geometrical diagram which is known as Venn diagram.


Example 
\(\begin{aligned} \xi&=\{\text{Amir, Hazura, Laila, Sandra,} \\&\quad\quad \text{Zamri, Dali, Pei San, Yana}\} \\\\A&=\{\text{Amir, Hazura, Laila} \\&\quad\quad \text{Sandra, Zamri}\}\\\\A'&=\{\text{Dali, Pei San, Yana}\} \end{aligned}\)




Subset of a set: 

Definition 
A set whereby all of its elements are the elements of another set.



 The symbol for subset is \(\subset\).
 ‘Is not a subset of’ can be denoted using the symbol \(\cancel{\subset}\).


Example 
Given the following sets.
\(\begin{aligned} Q&=\{x,y\} \\\\R&=\{v,w,x,y, z\} \end{aligned}\)
Is set \(Q\) is the subset of set \(R\)?

Yes, set \(Q\) is the subset of set \(R\) because every element of \(Q\) is found in \(R\).



 Empty set, \(\phi\) is a subset of any set.
 Set itself is a subset of any set.
 If a set contains \(n\) elements, then the possible number of subsets is \(2^n\).


Example 
List all the possible subsets for set \(\{k,l\}\).

We can see that the set contains \(2\) elements.
So, the possible number of subsets is
\(2^2=4\).
Thus, the possible subsets is
\(\phi,\{k\},\{l\},\{k,l\}\).



Represent subsets using Venn diagrams: 

 For an infinite set, its elements need not be written.


Example 
Given that,
\(\begin{aligned} A&=\{2, 4, 6, 8, 10,12, 14, 16, 18, 20\}\\\\B&=\{4, 8, 12, 16, 20\} \end{aligned}\)
The relationship of \(B\subset A\) can be represented using the Venn diagram as shown below.


