Complement Of Set

Definition: The complement of a set is the set that contains all of the universal set’s elements that aren’t found in the provided set. Assume A is a set of all coins, which is a subset of a universal set that includes all coins and notes; the complement of A is a set of notes (which do not include coins).

Example: If we’re talking about the total number of pupils at a school, we can break it down into two categories: boys and girls. Because boys and girls together make up the total number of pupils in that particular school, the number of boys is the complement of the number of girls.

We can say that the overall number of students is the universal set, that the number of boys is one subset of the universal set, and that the number of girls is the complement of the number of boys in mathematical terms.

So, in order to comprehend the concept of complement of sets, we must first grasp the concepts of sets, universal sets, and subsets.

What is the Complement of a Set?

If a universal set U has subset A, then the complement of set A is denoted by A’, is different from the elements of set A, as it includes the universal sets elements but not the elements of set A. A’ = x U: x A in this case. To put it another way, the complement of a set A is the difference between set A and the universal set.

Complement of Set Symbol: Any set’s complement is denoted by the letters A’, B’, C’, and so on. In other words, if the universal set (U) exists and a subset of the universal set (A) exists, the difference between the universal set (U) and the subset of the universal set (A) is the complement of the subset, i.e. A’ = U – A.

Example of Complement of a Set

The complement of set A is different from the components of A if the universal set includes all prime numbers up to 25 and set A = {2, 3, 7}.

Step 1 is to look for the universal set and the set for which the complement is needed. 

A = {2, 3, 7} 

U = {2,3,5,7,11,13,17,19,23}

Step 2 is to subtract (U – A)

Here , U – A = A’ 

= {2, 3, 5, 7, 11, 13, 17, 19, 23} – {2, 3, 7}

= {5, 11, 13, 17, 19, 23}

Types of Complement Of a Set

There are two kinds of set complements:

• Relative Complement
• Absolute Complement

Absolute Complement: The complement of A is just the set of all the elements contained in U that are not present in A. In the U, it is A’s relative complement.

Relative Complement: The set of all the items present in B but not in A is called the Relative Complement of set A with regard to set B. It’s also known as the “A-B” difference among two sets A and B. We’re only talking about the complements of sets A and B here, and it has nothing to do with universal sets.

Example

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

B = {3, 5, 7, 9, 11, 13, 15, 17,19, 21}

A-B = {2, 4, 6, 8,10}

B-A = {11, 13, 15, 17,19, 21}

Properties of Relative Complement: Relative Complement has two types of properties

• Complement Laws
• Difference Laws

Complement Laws: 

A-A = ∅

This demonstrates that A –A is always an empty set.

A-∅ = A

The set itself will always be A-∅.

∅-A = ∅

∅-A is the empty set at all times.

Difference Laws:

A-(∪C) = (A-B) ∩ (A-C)

A-(B∩C) = (A-B) ∪ (A-C)

Important Notes on Complement of a Set:

• The empty set, often known as the null set, is the universal set’s complement.
• The intersection set is made up of the components that are shared by both sets.
• A set that contains all of the elements from both sets A and B is called a union of two sets.

Conclusion

The complement of a set is the set that contains all of the universal set’s elements that aren’t found in the provided set. Assume A is a set of all coins, which is a subset of a universal set that includes all coins and notes; the complement of A is a set of notes (which do not include coins). If a universal set U has subset A, then the complement of set A is denoted by A’, is different from the elements of set A, as it includes the universal sets elements but not the elements of set A. A’ = x U: x A in this case. To put it another way, the complement of a set A is the difference between set A and the universal set. There are two kinds of set complements: Relative Complement & Absolute Complement.