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

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

A set that consists of all the elements under discussion.

  • The symbol for universal set is \(\xi\).
Complementary set:

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


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 \(P\).


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


The complement of set \(P\) is


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.

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

Subset of a set:

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 \(\cancel{\subset}\).

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 \(2^n\).

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 


Thus, the possible subsets is


Represent subsets using Venn diagrams:
  • For an infinite set, its elements need not be written.

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.