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In Maths, sets can be defined as a collection of well-defined objects or elements. A set can be represented by a capital letter symbol, and the number of elements in the finite set can be represented as the cardinal number of a set in a curly bracket {â€¦}.

For example, set A is a collection of all the natural numbers, such as A equals {1, 2, 3, 4, 5, 6, 7, 8, â€¦.. âˆž}.

Sets can be represented in three forms:

Roster Form: Example - Set of even numbers less than 8 = {2, 4, 6}.

Statement Form: Example: A = {Set of Odd numbers less than 9}.

Set Builder Form: Example: A = {x: x=2n, n âˆˆ N and 1 â‰¤ n â‰¤ 4}.

In this article, we are going to discuss the properties of the complement of a set, we are going to go through the properties of the complement of a set in brief.

A set has many types, such as;

Empty Set or Null Set: It has no element present in it. Example: A = {} is a null set.

Finite Set: It has a limited number of elements. Example: A = {1, 2, 3, 4}.

Infinite Set: It has an infinite number of elements. Example: A = {x: x is the set of all whole numbers}.

Equal Set: Two sets which have the same members. Example: A = {1, 2, 5} and B = {2 , 5, 1}: Set A = Set B.

Subsets: A set â€˜Aâ€™ is said to be a subset of B if each element of A is also an element of B. Example: A = {1, 2}, B = {1, 2, 3, 4}, then A âŠ† B.

Universal Set: A set that consists of all elements of other sets present in a Venn diagram. Example: A = {1, 2}, B = {2, 3}, The universal set here will be, U = {1, 2, 3}.

There are three properties of the complement of a set. Letâ€™s go through these three properties of the complement of a set:

Complement Laws: This is the first of the three properties of the complement of a set. The union of a set A and its complement denoted by Aâ€™ gives the universal set U of which A and Aâ€™ are a subset.

A âˆª Aâ€™ equals U

Also, the intersection of a set A and its complement Aâ€™ gives the empty set denoted by âˆ….

A âˆ© Aâ€™ = âˆ…

For Example: IfÂ U = {1 , 2 , 3 , 4 , 5 }Â andÂ A = {1 , 2 , 3} thenÂ Aâ€™ = {4 , 5}. From this, it can be seen that A âˆª Aâ€™ = U = { 1 , 2 , 3 , 4 , 5}

Also, A âˆ© Aâ€™ = âˆ…

Law of Double Complementation:

This is the second of the three properties of the complement of a set. According to this law of Double Complementation, if we take the complement of the complemented set named Aâ€™ then, we get the set A itself.

(Aâ€™ )â€™ equals A

In the previous example we can see that, if U = {1, 2, 3, 4, 5} and A = {1, 2, 3}Â then Aâ€™ = {4 , 5} . Now if take the complement of set Aâ€™ we get the following,

(Aâ€™ )â€™ = {1, 2, 3} = A , This gives us the set A itself.

Law of Empty Set and Universal Set:

According to this law, the complement of the universal set gives us the empty set and vice-versa that is,

âˆ…â€™ equals U And U equals âˆ…â€™

These three are the properties of the complement of a set. These properties of the complement of a set are useful in Mathematics.

Question 1) A universal set named U which consists of all the natural numbers which are multiples of the number 3, less than or equal to the number 20. Let set A be a subset of U which consists of all the even numbers and set B is also a subset of U consisting of all the prime numbers. Verify De Morgan Law.

Solution) We have to verify (A âˆª B)â€™ equals Aâ€™ âˆ© Bâ€™ and (A âˆ© B)â€™ equals Aâ€™âˆªBâ€™. Given that, Using the properties of the complement of a set, letâ€™s solve.

U equals {3, 6, 9, 12, 15, 18}

A equals {6, 12, 18}

B equals {3}

The union of both A and B can be given as,

A âˆª B equals {3, 6, 12, 18}

The complement of this union is given by,

(A âˆª B)â€™ equals {9, 15}

Also, the intersection and its complement are given by:

A âˆ© B = âˆ…

(A âˆ© B)â€™ equals {3, 6, 9, 12, 15,18}

Now, the complement of the set A and set B can be given as:

Aâ€™ = {3, 9, 15}

Bâ€™ = {6, 9, 12, 15, 18}

Taking the union of both these sets, we get,

Aâ€™âˆªBâ€™ = {3, 6, 9, 12, 15, 18}

And the intersection of the complemented sets can be given as,

Aâ€™ âˆ© Bâ€™ = {9, 15}

We can see that:

(A âˆª B)â€™ = Aâ€™ âˆ© Bâ€™ = {9, 15}

And also,

(A âˆ© B)â€™ = Aâ€™ âˆª Bâ€™ = {3, 6, 9, 12, 15,18}

Hence, the above-given result is true in general and is known as the De Morgan Law.

FAQ (Frequently Asked Questions)

Question 1: What is Set? Give 5 Examples.

Answer: Sets are usually symbolized either by uppercase, italicized, boldface letters such as A, B, S, or Z. Each number or object in a set is known as a member or element of the set. Examples include the set of all computers in the world, the set of all mangoes on a tree, and the set of all irrational numbers between 0 and 1.

Question 2: What is a Proper Set Example?

Answer: A proper subset of a set B is a subset of B that is not equal to B. In other words, you can say that if A is a proper subset of B, then all elements of A are in B but B contains at least one element that is not in A. For example, if A = {1, 3, 5} then B = {1, 5} is a proper subset of A.

Question 3: What are the Types of Sets?

Answer: **Types of a Set**

**Finite Set:**A set that contains a definite number of elements is called a finite set.Â**Infinite Set:**A set that contains an infinite number of elements is known as an infinite set.Subset

Proper Subset

Universal Set

Empty Set or Null Set

Singleton Set or Unit Set

Equal Set

Question 4: What is the Symbol for an Empty Set?

Answer: Symbol âˆ…

**Empty Set:** The empty set (or null set) is a set that has no members. Notation: The symbol âˆ… is used to represent the empty set, { }. Note: {âˆ…} does not symbolize the empty set; it represents a set that contains an empty set as an element and hence has a cardinality of one.