Types of sets

A set is a well-defined collection of distinct elements. Following are the types of a set.

Empty set

An empty set is a set that contains no elements and is also known as a null or void set. It is denoted by $\phi$ or $\{\}.$

Example

Let $A$ be the set of odd numbers divisible by 2. Since there are no elements that satisfy the criteria of being both odd and divisible by 2, it is an empty set: $A=\{\}$

Singleton set

A singleton set is a set that contains only one element.

Example

$A=\{2\}$ is a singleton set as it has only one element.

Finite set

A finite set is a set with a countable number of elements.

Example

$X = \{x:x \text{ is a vowel in the English alphabet}\}$

Infinite set

An infinite set has an uncountable number of elements.

Example

$X = \{x:x\text{ is an even number}\}$

Equal sets

Two sets are said to be equal if they have the same elements, regardless of the order.

Example

Let $A = \{1,2,3\}$ and $B = \{2,3,1\}$. Since the elements are the same, $A$ and $B$ are equal sets. This equality is denoted by the expression: $A = B$

Equivalent sets

Two sets $A$ and $B$ are equivalent if they contain the same number of elements.

Example

For sets, $A = \{1, 3, 5, 7, 9\}$ and $B = \{a, e, i, o, u\}$

As set $A$ has five elements and set $B$ also has five elements. Therefore, sets $A$ and $B$ are equivalent. This is denoted by the expression: $A \equiv B$

Universal set

A universal set is a set that contains all the elements of other sets. It is represented as $U$.

Example

Let $A=\{1, 2, 3\}$ and $B=\{2, 4, 6\}$. The universal set will be, $U = \{1, 2, 3, 4, 6\}$

Disjoint sets

Two sets $A$ and $B$ are disjoint if they have no elements in common. It is represented as $A \cap B = \phi$

Example

$A = \{1, 3, 5, 7\}$ and $B = \{2, 4, 6, 8\}$ are disjoint sets because there is no common element between them: $A \cap B = \phi$

Power set

The power set of A, denoted by $P(A)$, is the set of all possible subsets of $A$. It includes the empty set and the set itself.

Example

Let $A = \{1,2\}$. The power set of $A$will be $P(A) = \{\phi, \{1\}, \{2\}, \{1, 2\}\}$.

Note: The total number of subsets formed for any set $A$ containing $n$ elements is $2^n$. Thus, $P(A)$ has $2^n$ elements.

Subset

A set $X$ is considered a subset of another set $Y$ if every element of $X$ is also an element of $Y$. $X$ and $Y$ may be equal. It is denoted by $X \subseteq  Y$

Example

Set $X = \{1, 2\}$ is a subset of set $Y = \{1, 2, 3\}$: $X \subseteq Y$

Proper subset

Set $A$ is a proper subset of another set $B$ if $A$ is a subset of $B$, but $A$ is not equal to $B$. It is denoted by $A \subset  B$

Example

Set $A = \{1, 2\}$ is a proper subset of set $B = \{1, 2, 3\}$: $A \subset  B$.

Superset

Set $A$ is the superset of set $B$ if all the elements of set $B$ are the elements of set $A$. It is represented as $A \supset  B$

Example

Set $A = \{1, 2, 3, 4\}$ is a superset of set $B = \{2, 3, 4\}$: $A  \supset B$.