Empty Set

Sets are the well-defined and organised collection of all objects and elements. A set can comprise any element, let it numbers, days of a week, cards in a deck, etc. They are used to represent and differentiate one element from another. 

One basic rule of a set is that it does not change from person to person. We usually use capital letters to represent sets and enclose their contents within curly brackets.

Example

Let us consider a simple example of a set.

Let A be a set that comprises the numerals 10 to 15.

Then, we can give it as,

A = {10,11,12,13,14,15}

The numerals from 10 to 15 are said to be the elements of the set. They do not have any proper order but should not be repetitive. If we use alphabets within the elements, they need to be small letters.

We can denote the elements in the set in the following way:

10 ∈ A

It means that element 10 belongs to set A.

Now, 

16 ∉ A 

It means that element 16 does not belong to set A.

Commonly known sets are:

• N: Set of all natural numbers
• Z: Set of all integers
• Q: Set of all rational numbers
• R: Set of all real numbers
• Z+: Set of all positive integers

Types of Sets

Let us look at the various sets.

• Finite set

This type of set consists of a limited or finite number of elements. For example, set B contains numbers from 1 to 5. Therefore:

B = {1,2,3,4,5}

Another example is a set of days in a week.

• Infinite set

This type of set consists of an unlimited number of elements that are not countable. As a result, these sets cannot be represented wholly.

Sets of all integers, sets of all rational numbers, etc., are illustrations of this kind.

• Empty sets

Empty sets are the type of sets that are empty and contain no elements. Being referred to as void sets, they are designed for simple calculation. These sets are unique and are useful in cases where there are no indications of specific answers.

Examples:

• A month with 32 days
• A week with two Sundays
• An animal with five legs
• A car with ten doors

Let us understand empty sets in detail.

Representation of Empty Sets

The special character {} can represent empty sets, and their contents are enclosed within curly brackets like every other type of set.

The notation for empty sets are:

A = {}

To understand this concept better, let us consider an example of an empty set, denoted as A containing multiples of 4 between 5 and 7. Since there are no multiples of 4 between 5 and 7, the given set can be considered empty. Therefore:

Set A = {y: y is a multiple of 5 and 5<y<7}

Basic Properties of Empty Sets

Here are the properties of empty sets.

• Subset

We can regard empty sets as the subsets for any given set under consideration. It can be represented by:

 φ ⊆ A; ∀ A

• Union with empty sets

A union operation between any set and an empty set will always provide the set itself as a result. It can be represented by:

A ⋃ φ = A; ∀ A

• Intersection with Empty Sets

An intersection operation between any set with an empty set will always provide the set itself as a result. It can be represented by:

A ⋂ φ = φ; ∀ A

• Cardinality

The term cardinality represents the size of the given set. It provides the total number of elements present in a set. Therefore, the cardinality of an empty set is zero. It can be represented by:

|φ| = 0

• Cartesian Products

The Cartesian product between any set with an empty set will always provide an empty set as the result. It can be represented by:

A × φ = φ; ∀ A

Conclusion 

Sets may contain any kind of elements, such as natural numbers, integers, rational numbers, and real numbers. Sets containing no elements are said to be empty sets or void sets. The basic properties of empty sets include being a subset, forming unions and intersections with other sets, cardinality, and Cartesian products. 

Venn diagrams represent the relationships between different types of sets. These study material notes on empty sets can help us better understand this popular concept in mathematics. It is helpful for students preparing for competitive exams.