Set notation and Venn diagrams 2

Last updated: September 24, 2021

What are set notations?

Set notations are used to determine if the elements are contained in the superset or not. For example, if $$X=\left \{ a, e, i, o, u \right \}$$, $$a\in X$$ and $$b\notin X$$.

Subset of a set

Sets and subsets are collections of elements. A set contains some elements and some of those elements are contained in another set which is called the subset of the main set. It is denoted as $$A\subseteq B$$.

Proper subset of a set

If every element of the set $$A$$ is an element of the set $$B$$ and $$B$$ contains at least one element which does not belong to $$A$$, that is, if $$A\subset B$$ and $$A\neq B$$, we can say that $$A$$ is a proper subset of $$B$$ and it is denoted by $$A\subset B$$.

For example, $$A=\left \{2,3,4,6 \right \}$$ is a proper subset of $$B=\left \{4,3,2,6,5,8 \right \}$$.

Equal sets

If set $$A$$ and $$B$$ have same and equal elements, the sets $$A$$ and $$B$$ are called equal sets. It is denoted as $$A=B$$.

Empty set

A set consisting of no points is called an empty set or null set. It is denoted by $$\Phi$$ or $$\left \{ \right \}$$.

Singleton set

A set consisting of a single element is called a singleton set. For example, $$\left \{1 \right \}$$, $$\left \{2 \right \}$$, $$\left \{a\right \}$$ etc. are singleton sets.

Universal set

The set that contains all elements under consideration is known as the universal set. It is denoted as $$U$$.

Power set

If a set contains all the elements and subsets of a set $$A$$, the set is called the power set. It is denoted as $$P\left (A \right )$$.

Union of sets

The union of two sets $$A$$ and $$B$$ is written as $$A\cup B$$. It indicates the set of elements that belongs to $$A$$ or $$B$$ or both.

In the diagram, the shaded portion represents  $$A\cup B$$.

E1.2: Use language, notation and Venn diagrams to describe sets and represent relationships between sets

In this article, we shall use the set of natural numbers, the set of integers, the set of real numbers, the set of rational numbers and the set of complex numbers. Some operations discussed above will be used to solve the examples.

Worked examples on set notation

Example 1: Consider the three sets $$P={ 0,1,4,7,9,6,3} , Q={ 9,6,3}$$ and $$R={ 0,1,4,7}$$.

Find the $$P\cup Q$$ and $$Q\cap R$$

Step 1: Write the given values.

$$P=\left \{ 0,1,4,7,9,6,3 \right \} , Q=\left \{ 9,6,3 \right \}$$ and $$R=\left \{ 0,1,4,7 \right \}$$

Step 2: Find $$P\cup Q$$ using the sets $$P$$ and $$Q$$.

Write all the elements that are members of sets $$P$$ and $$Q$$.

$$P\cup Q=\left \{ 0,1,4,7,9,6,3 \right \}$$

Step 3: Find the $$Q\cap R$$ using the sets $$Q$$ and $$R$$.

Write all the elements that are common in sets $$Q$$ and $$R$$.

$$Q\cap R =\left \{ \right \}$$

The result is an empty set.

Example 2: Consider the two sets $$A=\left \{ p,q,r,s,t \right \}$$ and $$B=\left \{ a, b, p, q, r \right \}$$. Find $$A\cap B$$.

Step 1: Write the given values.

$$A=\left \{ p,q,r,s,t \right \}$$ and $$B=\left \{ a, b, p, q, r \right \}$$.

Step 2: Find $$A\cap B$$ using the sets $$A$$ and $$B$$.

Write all the elements that are common in sets $$A$$ and $$B$$.

$$A\cap B=\left \{ p, q, r \right \}$$.