Types of sets

In set theory, there are various set types. All the operations of set theory are based on sets. Set ought to be a collection of individual terms in a domain.

The universal set embodies each and every element of a domain. In this article, we will look at various types of sets.

There are various types of sets in set theory. They are presented below:

Empty set

Singleton set

Finite and Infinite set

Union of sets

Intersection of sets

Difference of sets

Subset of a set

Disjoint sets

Equality of two sets

Empty Set

An empty set is a set that possess no element. An empty set is as well known as Null set and Void set. Number of element in set X is denoted by n(X).

The empty set is represented with the symbol Φ. Therefore, n(Φ) = 0. The cardinality of an empty set is zero because it has no element.

In mathematics, empty set is a set theory related topic. A set without any elements is said to be an empty set.

This article helps you understand empty set by providing a clear idea about empty set with a few sample problems. The other name of empty set is null set ϕ.

Consider two sets X = {a, b, c, d} and Y = {1, 2, 3, 4, 5, 6}. Consider another set Z which represents the intersection of X and Y. There is no common element for the set X and Y. Therefore, intersection of X and Y is null.

Z = { }

The representation of empty set is { }.

Cardinality of Empty Set:

Since we know that the cardinal number represents the number of elements that are available in the set and by the definition of an empty set, we are aware that there is no element in the empty set.

Therefore, the cardinal number or cardinality of an empty is zero.

Properties of Preparation for Empty Set:

1. Empty set is taken as subset of all sets. ϕ⊂X

2. Union of empty set ϕ with a set X is X. A∪ϕ=A

3. Intersection of an empty set with a set X is an empty set.

Solved Examples Question 1:

A is a set of alphabets and B is a set of numbers. What is the intersection of A and B?

Solution:

A ∩ B = { }

Question 2:

Write the set A which is a set of goats with 10 legs.

Solution:

A = { }

Power Set of the Empty Set

A set is called the power set of any set, if it is made up of all subsets of that set. We can use the notation P(S) for representing any power set of the set.

Now, from the definition of an empty set, it is evident that there is no element in it and therefore, the power set of an empty set i.e. P(ϕ) is the set which contain only one empty set, thus P(ϕ) = {ϕ}

Cartesian Product Empty Set

The Cartesian product of any two sets say A and B are denoted by A X B. There are a few conditions for Cartesian product of empty sets as represented below:

If we have two sets A and B in such a manner that both the sets are empty sets, then A X B = ϕ x ϕ = ϕ. It is clear that, the Cartesian product of two empty sets is again an empty set.

If A is an empty set and B = {1, 2, 3}, then the Cartesian product of A and B is shown as:

A X B = {ϕ} . {1,2,3} = {ϕ X 1, ϕ X 2, ϕ x 3}

= {ϕ, ϕ, ϕ}

= {ϕ}

Therefore, we say that if one of the set is an empty set from the given two sets, then again the Cartesian product of these two sets is an empty set.

Examples of Empty Sets

Given below are some of the examples of empty sets.

Solved Examples

Question 1:

Which of the following represents the empty set?

1. A set of cats with 4 legs

2. A set of apples with red color

3. A set of positive numbers in which all are less than 1

4. A set of rectangles with 4 sides

Solution:

Option 1: A set of cats with 4 legs.

This set is possible where cats are having 4 legs.

Option 2: A set of apples with red color

This set is possible where apple is in red color.

Option 3: A set of positive numbers in which all are less than 1.

This set is not possible because the positive numbers must be greater than 1. So, this set is said to be an empty set.

Answer: The set in option 3

Question 2:

A is a set of numbers from 1 to 10, B is a set of negative numbers. What is the intersection of A and B?

Solution:

Given:

A = set of number from 1 to 10.

= {1, 2, 3, 4, 5, 6, 7, 8, 9, and 10}

B = set of negative numbers

= {-1, -2, -3, -4,….}

Intersection of A and B = A ∩ B

= { }

Answer: The intersection of the given sets is an empty set.

Singleton Set

A set that has one and only one element is referred to as Singleton set. It is as well occasionally referred to as a unit set.

The cardinality of singleton set is one since it is made up of only one element.

If A is a singleton set, then we can express the set as A = {x : x = A}

Example: Set A = {5} is a singleton set.

Finite and Infinite Set

A set that has preset number of elements or finite number of elements is known as Finite set. Example {1, 2, 3, 4, 5, 6} is a finite set and the cardinality is 6, because it has 6 elements.

A set that does not have a preset number of elements is known as infinite set. It may be uncountable or countable. The unions of a few infinite sets are infinite and the power set of any infinite set is infinite.

Examples:

1. Set of all the days in a week is a finite set.

2. Set of all integers is infinite set.

Union of Sets

Union of two or else majority of the numbers of sets could be the set of all elements that belongs to every element of all sets. In the union set of two sets, every element is written only once even when the two sets contain the element.

The union of set is represented by the symbol ‘∪’. Given sets A and B, then the union of these two sets is A U B and is pronounced as A union B.

Mathematically, we can represent union of sets as A U B = {x: x ∈ A or x∈ B}

The union of two sets is always commutative i.e. A U B = B U A.

Example: A = {1,2,3}

B = {1,4,5}

A ∪ B = {1,2,3,4,5}

Intersection of Sets

Intersection is an operation on sets. The intersection of sets is the set of elements that are common in the two. It is merely the opposite of union of set.

It is a very significant and vital concept in set theory. Before we learn about intersection, we ought to understand a few basic concepts like what is set.

A set is a well defined group of data. It's data is referred to as it's members or elements. We represent the set by capital letters A, B, C, X, Y, Z, etc.

We make use of the concept of set in day to day life. For instance, a team that is made up of five members is a set.

Difference of Sets

The difference of set A to B is denoted as A - B. That is, the set of element that are in set A which are not in set B is

A - B = {x: x ∈ A and x ∉ B}

And, B - A is the set of all elements of the set B which are in B but not in A i.e.

B - A = {x: x ∈ B and x ∉ A}, mathematically.

Example:

If A = {1,2,3,4,5} and B = {2,4,6,7,8}, then

A - B = {1,3,5} and B - A = {6,7,8}

Subset of a Set

In set theory, a set P is the subset of any set Q, if the set P is enclosed in set Q. It means all the elements of the set P that as well belongs to the set Q. It is represented as '⊆’ or P ⊆ Q.

Example:

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

B = {1,2,3,4,5,7,8}

Here, A is said to be the subset of B.

Disjoint Sets

If two sets A and B have no common elements or we can say that the intersection of any two sets A and B is the empty set, then these sets are referred to as disjoint sets i.e.

A ∩ B = ϕ. This means that when n (A ∩ B) = 0, then the sets are referred to as disjoint sets.

Example:

A = {1,2,3}

B = {4,5}

n (A ∩ B) = 0.

Therefore, these sets A and B are disjoint sets.

Equality of Two Sets

Two sets are said to be equal or identical to each other, if they enclose the same elements. When the sets P and Q is said to be equal, if P ⊆ Q and Q ⊆ P, then we will write as P = Q.

Examples:

1. If A = {1, 2, 3} and B = {1, 2, 3}, then A = B.

2. Let P = {a, e, i, o, u} and B = {a, e, i, o, u, v}, then P ≠ Q, since set Q has element v as the extra element.

Disjoint set: Two sets A and B are said to be disjoint if A $\cap$ B = $\phi$

A few set questions and solutions

Question 1:

If A = {1,3,4,6,9} and B = {2,4,6,8}, find A ∩ B. What is your conclusion?

Solution:

We have given that A = {1, 3, 4, 6, 9} and B = {2, 4, 6, 8}

To find the intersection of A and B,

thus, A ∩ B = {1, 3, 4, 6, 9} ∩ {2, 4, 6, 8}

A ∩ B = {4, 6}

Question 2:

If A = {1,3,5,7,9} and B = {2,4,6,8}, find A ∩ B. What is your conclusion?

Solution:

We have A ∩ B = {1,3,5,7,9} ∩ {2,4,6,8}

If no element match in the two sets, both the sets are said to be disjoint sets.

Therefore, A and B are disjoint sets. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you