# Introduction to sets and subsets

Last updated: September 24, 2021

## What are sets?

A set is a collection of well defined objects. The objects in a set are called the members or elements of the set, and a certain condition defines their membership. The elements of a set can be anything that satisfies the definition of membership.

Suppose $$S$$ be the set that is a collection of objects. And $$x$$ is an object that belongs to $$S$$, i.e., member of $$S$$. It is written as $$\{S=x:Q(x)\}$$. It means $$S$$ is the set of objects for which the statement $$Q(x)$$ involves $$x$$ is true. Sets are generally denoted by capital letters $$A, B, C, P, Q, X, Y$$ etc. Elements of the set are denoted by small letters $$a, b, c, p, q, x, y$$ etc. If $$x$$ is not the member of the set $$S$$, then it is written as $$x\notin S$$ and read as $$x$$ does not belong to $$S$$.

Subset: If every element of a set $$A$$ is an element of a set $$B$$, then $$A$$ is called a subset of $$B$$. It is denoted as $$A\subseteq B$$.

Real-life application of sets: humans, animals, plans, things etc. are sets, which belong to the Earth. So, these humans, animals, plants are a subset of Earth

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

The operations of the set are union and intersection. The union of two sets, $$A$$ and $$B$$, is written as $$A\cupB$$. It means the set of points belonging to one of the sets, $$A$$ and $$B$$, belongs to $$A$$ or $$B$$ or both.

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

For example, $$A=\{1,2\}$$ and $$B=\{3,4,5\}$$ then the union of $$A$$ and $$B$$ is the combination of elements in both sets, i.e., $$A\cup B=\{1,2,3,4,5\}$$

Therefore, $$A\cup B=\{x:x\in A \;\text{or}\; x\in B \;\text{or}\; x\in \;\text{both A and B}\}$$

The intersection of two sets $$A$$ and $$B$$ is written as $$A\cap B$$. It means the set of points belongs to both $$A$$ and $$B$$.

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

Therefore, $$A\cap B=\{x:x\;\in A \;\text{and}\; x\in B\}$$

For example, $$A=\{1,2\}$$ and $$B=\{2,3,4\}$$, then the intersection of $$A$$ and $$B$$ is $$A\cap B=\{1,2\}$$

If $$A\cap B=\Phi$$, it means there is no common element in $$A$$ and $$B$$. In this case, the set $$A$$ and $$B$$ are said to be disjoint.

Sometimes, $$A+B$$ can be written in place of $$A\cup B$$. Sometimes $$A-B$$ can be written in place of $$A\cap B$$.

### Worked examples of sets

Example 1: If $$A=\{4,5,6\}$$ and $$B=\{1,8,9\}$$, find $$A\cup B$$. And show in the Venn diagram.

Step 1: Write the given values.

$$A=\{4,5,6\}$$ and $$B=\{1,8,9\}$$

Step 2: Do the union operation.

$$A\cup B=\{1,4,5,6,8,9\}$$

Therefore, the union of the $$A\cup B$$ is $$\{1,4,5,6,8,9\}$$.

The Venn diagram is as shown below

Example 2: If $$A=\{1,2,4,5\}$$ and $$B=\{1,2,6,7\}$$, find $$A\cap B$$. And show in the Venn diagram.

Step 1: Write the given values.

$$A=\{1,2,4,5\}$$ and $$B=\{1,2,6,7\}$$

Step 2: Do the intersection operation.

$$A\cap B=\{1,2\}$$

Therefore, the intersection of $$A\cap B$$ is $$\{1,2\}$$.

The Venn diagram is as shown below.

Example 3: If $$P=\{1,2,3,4\}$$ and $$Q=\{1,2\}$$, find $$A\cup B$$. And show in the Venn diagram.

Step 1: Write the given values.

$$P=\{1,2,3,4\}$$ and $$Q=\{1,2\}$$.

Step 2: Do the intersection operation.

$$A\cup B=\{1,2,3,4\}$$.

The result of $$A\cup B$$ is the same as the set $$P$$.

Therefore, $$Q$$ is the subset of $$P$$.

Hence, $$A\cup B$$ is $$\{1,2,3,4\}$$.

The Venn diagram is as shown below