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

 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.