Empty Sets are sets with no items or elements in them. They are also called null sets. The symbol (phi) ∅represents the empty set and is written as ∅ = { }. It is also known as a void set or a null set. When compared to other sets, empty sets are seen to be distinctive.
Empty sets are used to simplify computations and are most commonly employed when categorizing odd or unusual items.
What is an Empty Set?
The empty set, null set, or void set is a set that does not include any elements. For example, the collection of possibilities for rolling a die and obtaining a number larger than 6. As we all know, the results of a dice roll are 1, 2, 3, 4, 5, and 6. As a result, the set containing integers larger than 6 here will be. This indicates that there will be no elements and is referred to as the empty set.
Empty Set Definition
The empty set is a set that contains no elements. It is a unique set that is defined by the absence of any members.
Empty Set Symbol
An empty set, null set, or void set is a set that has no elements and is symbolized by the symbolÂ
∅ (phi)
OR
{ Â }
When two dice are rolled simultaneously, the collection of the numbers for threesomes gets a number larger than 25. As a result, the set containing integers bigger than 25 here will be zero, i.e. This implies that there will be no items and that the resulting set will be empty.
Non-Empty Sets
As we know, the collection of elements is set and any set with at least one element in it is called a non-empty set. In other words, any set with at least one element is called a non-empty set. Some examples of non-empty sets are:
- {1, 2, 3} is a non-empty set because it contains three elements: 1, 2, and 3.
- {“apple”, “banana”, “orange”} is also a non-empty set with three elements.
- The set {5} is non-empty since it contains the single element 5.
- The set of all positive integers greater than 2 can be represented as {3, 4, 5, 6, 7, …}, and it is a non-empty set.
Examples
- Set A = {x: x is a prime number between 32 and 36}, as there are no prime numbers between 32 and 36 set A is an empty set. Set A = ∅.
- Set B = {x: x are numbers greater than 6 while rolling a die}, there is no number greater than six in a die, so set B is an empty set. Set B = ∅.
- Set C = {x: x are natural numbers that satisfy x2 + x + 1 = 0}, there is no such natural number that satisfies the given condition, so set C is an empty set. Set C = ∅.
Properties of Empty Set
Property 1: Empty Set is a Subset of Every Set.
According to the property, the empty or null can be regarded as a subset of any set. That is, given a set P, the empty set is a subset of P, such that ∅ ⊆ P; ∀ P. Let’s look at an example to better understand the attribute.
Example:
Assume P = {4, 12, 21} is a finite set.
All possible subsets of P are,
P = ∅, {4}, {12}, {21}, {4, 12}, {12, 21}, {4, 21}
This demonstrates that for any finite or infinite set, if all potential subsets are considered, an empty set will always be included.
Property 2: Union with Empty Set
The property asserts that the union of any set with an empty set always results in the set itself. Q ⋃ ∅ = Q; ∀ Q. is a mathematical expression.
Example:
If B = {7, 8, 9, 10} then:
B U ∅ = {7, 8, 9, 10} U { }
B U ∅ = {7, 8, 9, 10}
This attribute is also valid since an empty/null set contains no elements and its union with any other set yields the same as the end result.
Property 3: Intersection with Empty Set
This property, in contrast to the union property, states that the intersection of any given set’s sets with an empty one will always result in an empty set. E â‹‚ ∅ = ∅; ∀ E is the mathematical expression.
Example:
If we are given E = {10, 20, 30, 40}
we may write: E ∩ ∅ = {10, 20, 30, 40} ∩ {}
E ∩ ∅=∅
This characterisolds is true because the empty/null set has no items, implying that no common component exists between the provided set and the empty one.
Property 4: Cardinality of Empty Set
Cardinality specifies the total number of items in a given set. As we know, an empty kind of set has no items and so has 0 cardinality. This is represented as |∅| = 0.
Example:
Consider the set X = {x:Â x is an odd multiple of 2}
Odd numbers are those that cannot be divided by two. As a result, there are no odd multiples of 2. There is no element in either set A or set A = {}. As a result, the cardinality of set A =Â 0.
Property 5: Power Set of Empty Set
An empty set has no items. As a result, the power set of an empty set{}Â may be defined as P(E) ={}.
- An empty or null set’s power set contains exactly:
- A set that includes a null set.
- It contains zero or null items.
- The sole subset is the empty set.
- As a result, the number of power set components is 20 = 1
- As a result, the power set only has one element, which is the empty set itself.
Property 6: Cartesian Product of Empty Set
The multiplication of the provided sets is defined by the Cartesian product. Taking the cartesian product of any set with an empty/null set always yields an empty result. The explanation is obvious: the empty set includes no items. Mathematically expressed as: B x ∅ = ∅; ∀ B
Example:
If we are given with E = {10, 20, 30, 40} then:
E x ∅ = {10, 20, 30, 40} x {}=∅; ∀ E
Property 7: Complement of Empty Set
The universal set is the complement of a set that is an empty/null set. The ultimate solution is a universal set itself if we carry the difference between a universal set and an empty type set. ‘∅’ = U is the mathematical notation. This is because the universal set contains all items, but the empty set has none.
Property 8: Empty Set Subset
We may think of an empty set as a container with nothing in it. The empty set exists in the same way as the container does. An empty set’s lone subset is the set itself, i.e. P ⊆ ∅ ⇒ P = ∅.
Difference Between Zero Set and Empty Set
|
Zero Set
|
Empty Set
|
| A zero set is a set in which zero is the alone element. |
An empty has has no elements. |
| It is represented by the number {0}. |
It is represented as {}. |
|
Example,Â
Suppose P = {0}
The set symbolizes a zero set that contains just zero.
|
Example,Â
A set Q = {Â x: x is a prime number between 20<x<23}
The numbers that run from 20 to 23 are 21, 22. Because both of them are not prime, set Q is empty.
Q = { } OR  Q = ∅
|
Representation
Empty sets are represented in set theory by the empty curly brackets {} that are commonly used to designate sets. However, because empty sets are distinct types of sets, they can also be represented by a special character ∅. Consider the case of an empty set A that contains multiples of 5 between 6 and 8. The supplied set is empty since there are no multiples of 5 between 6 and 8. Set A = {y: y is a multiple of 5 and 6<y<8.}
- This empty set is denoted by the symbol A = {}
- The same empty set A may alternatively be written using the notations: Empty set = { } or, X = { }.
- To symbolize an empty set, we can use a special symbol ∅.
Venn Diagram – Empty Set
Venn Diagrams are often regarded as the best tool for representing connections between sets, particularly finite sets. An empty set can be represented using a Venn diagram. The following diagram depicts the relationship:
Consider the following sets: A = {1, 3, 5}Â and B = {2, 4, 6}
Because there are no common items that are intersecting elements between the two sets X and Y, we may deduce that the intersection between these two sets is empty. Consequently, A ∩ B = ∅.
Solved Examples on Empty Set
Example 1: Consider the following set A = {11, 12, 13, 14, 15, 16}. Determine the union of this set A and an empty set.
There are no elements in an empty set. The union of set A with the empty set is seen below:
A U ∅ = {11, 12, 13, 14, 15, 16} U { }
A U ∅ = {11, 12, 13, 14, 15, 16}
This demonstrates the notion that every set’s union with an empty set is the set itself.
Example 2: Find the cardinality of set X where set X = {x: x is an odd multiple of 12}.
To begin solving this problem, we shall simplify the set.
Because there are no odd multiples of 12, the set is empty.
Cardinality may be determined as follows:
|∅| = {x: x is an even multiple of 12}
|∅| = 0
Example 3: Find the complement of the set of even natural numbers less than 10.
First, we form the set A.
Because there are 4 even natural numbers less than 10 , the set is A = {2,4,6,8}.
Cardinality may be determined as follows:
|∅| = {x: x is an even natural number less than 10}
|∅| = 4
A’ = U – A = {1,3,5,7,9}
|∅’| = 5
Summary – Empty Set
An empty set is a set that doesn’t contain any element. For example, a set of even numbers lesser than 2 is an empty set. Because its cardinality is specified and equal to zero, an empty set is a finite set. The symbol (phi) ∅represents the empty set and is written as ∅ = { }. Except for the empty set, every other set is a proper subset of the empty set, and no set is a proper subset of itself. The empty set, null set, or void set is a set that does not include any elements.
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