Sets and subsets

Set and subset are a collection of elements. Set contains elements, and if some of those elements are contained in another set, then the second set is called the subset of the main set. It is denoted as $$A\subseteq B$$.

For example, education contains all subjects. So, maths, physics, chemistry, English are a subset of education.

Now discuss the different types of sets that are subset, superset, equality set, proper set, universal set, finite set, infinite set, power set, null set, index set, singleton set, pairwise disjoint sets, etc.

Superset

If $$A$$ is a subset of $$B$$, it means $$A$$ is contained in $$B$$. We can also say that $$B$$ is a superset of $$A$$. It can be written as $$A\supset B$$.

Equality of sets

If two sets, $$A$$ and $$B$$, are equal, it is written as $$A=B$$. Every element of $$A$$ belongs to $$B$$, and every element of $$B$$ belongs to $$A$$ if and only if $$ x\in A\Leftrightarrow x\in B$$ or $$A\subset B\;\text{and}\;B\subset A\Leftrightarrow A=B$$

Proper set

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

For example: $$\{2,3,4,6\}$$ is proper subset of $$\{4,3,2,6,5,8\}$$.

Universal set

Consider all the groups to be the subsets of a given fixed set known as universal of discourse. It is denoted by $$U$$ or $$X$$.

Infinite set: It is called an infinite set because it consists of an endless number of elements.

For example, a set of natural numbers $$N$$, a set of rational numbers $$R$$

Power set

The power set of a set is defined as the family consisting of all subsets of a set. It is denoted by $$P$$.

Null set

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

Indexed set and index set

Let $$S_{t}$$ be a non-empty set for each $$t$$ in a set $$\bigtriangleup$$. These sets $$ A_{1},A_{2},A_{3},\cdots , A_{n}$$ are called indexed set, and the set $$\Delta =\{1,2,3,\cdots, n\}$$ is called index set. The suffix $$t\in \Delta$$ of $$ A_{t}$$ is called an index set, a family of sets is denoted by $$\{A_{t}:t\in\Delta \}$$ or $$\{A_{t}\},t\in\Delta$$.

Singleton set

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

Pairwise disjoint sets

A family $$ {A_{n}}$$ of sets is said to be pairwise disjoint if $$A_{n}\cap A_{s}=\Phi$$, for all $$ t,s\in \Delta$$ and $$ t\neq s$$.

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

Now, let us discuss the set operations. Set operations are union and intersection. We use these operations to find the set is either a proper subset or not. And it is denoted by some specific symbols. If set $$A$$ is a proper subset of $$B$$, it can be denoted by $$ A\subset B$$. If set $$A$$ is a subset of $$B$$, then it can be denoted by $$ A\subseteq B$$. If the set is not a subset, it is denoted by $$ A\nsubseteqB$$.

Worked examples of sets and subsets

Example 1: If $$A=\{1,3,5\}$$, $$B=\{1,2,3,4,5\}$$ and $$C=\{2,8\}$$, find the relation between the sets.

Step 1: Write the given values.

$$A=\{1,3,5\} , B=\{1,2,3,4,5\}$$ and $$C=\{2,8\}$$

Step 2: Take $$A$$ and $$B$$, and do the intersection operation.

$$A\cap B= \{1,3,5 \}$$

Hence, $$A\cup B$$ is $$\{1,2,3,4,5\}$$.

Step 3: Identify the relation between the sets.

$$A\cap B=\{ 1,3,5 \}$$

The result is the same as set $$A$$.

So, $$A$$ is a proper subset of $$B$$, i.e., $$A\subset B$$.

Step 4: Take set $$C$$ and $$B$$, and do intersection operation.

$$B\cap C=\{2 \}$$

Hence, $$B\cap C$$ is $$\{2\}$$.

Step 5: Identify the relation between the sets.

$$B\cap C=\{2\}$$

The intersection of $$B$$ and $$C$$ is only one element.

So, $$C$$ is not a subset of $$B$$.

It is clearly shown in the Venn diagram.

Example 2: If $$X=\{1,2,3,4\}$$ and $$Y=\{5,6,7,8\}$$, find the $$X\cap Y$$.

Step 1: Write the given values.

$$X=\{1,2,3,4\}$$ and $$Y=\{5,6,7,8\}$$

Step 2: Take $$X$$ and $$Y$$, and do the intersection operation.

$$X\cap Y=\{ \}$$

Hence, there is no common element in both sets. Therefore, it is a null set.