Complement of Sets

What are sets?

Sets are a well-defined collection of objects or elements. A Universal Set is any set that contains all of the items or elements linked to a specific context and is symbolised by the letter U.

The Complement of Set A that is a subset of the Universal Set U, contains the elements that are members of the Universal Set but not from Set A. Set A’s complement is indicated by A′.

Complement of Sets

Let us now look at the definition of the complement of sets:

If a universal set (U) has a subset A, then the complement of set A, denoted by A’, is different from the elements of set A, as it includes the universal set’s 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 a universal set and set A  .

Symbol of Complement of sets:

A’, B’, C’, and so on denote the complement of sets. To put it another way, if the universal set (U) is given and the subset of the universal set (A) is given, the difference between the universal set (U) and the subset of the universal set (A) is the complement of sets, i.e., A’ = U – A.

Let us know about the complement of sets in a better way with the help of the Venn diagram:

The above complement of set A that is A’ can be seen. A’ does not belong to set A, and set A does not belong to A’. Both A and A’ are, however, subsets of U.

Properties of Complement of sets

There are four properties of complements of sets. They are as follows:

1. Complement Laws- 

If A is a subset of the universal set, then A’ is likewise a subset of the universal set; thus, the universal set is the union of A and A’, denoted by A∪ A’ = U.

The intersection of Sets A and A’ yields the empty set ∅”, which may be written as A ∩ A’ = ∅

Take this for example,

If U = {3,4,5,6,7}, A ={ 6,7} and B = {3,4}

A’ = { 3,4,5} and B’ = { 5,6,7}

A∪ A’= U = {3,4,5,6,7}

Also, A ∩ A’ = ∅

• Law of Double Complementation-

The original set is the complement of the supplemented set in this law, (A’)’ = A.

The double complement of A is thus the complement of the set A′, where A′ is the complement of A.

Taking the previous example of U = {3,4,5,6,7}, A ={ 6,7} and B = {3,4}, 

The complement of (A’)’ = {6,7} which is the same as set A.

• Law of Empty Set and Universal Set-

The universal set’s complement is an empty set, commonly known as a null set (∅) and the empty set’s complement is the universal set.

Because the universal set contains all items and the empty set contains none, their complements are exactly opposite of each other, as shown by the symbols 

∅’= U and U’ = ∅.

Again taking the above U={3,4,5,6,7} example, that includes all elements of both set A as well as set B like the universal set, therefore:

U’ = ∅(null set) and ∅’ = {3,4,5,6,7}

• De Morgan’s Law-

The complement of two sets united is equal to the complement of two sets intersected. De Morgan’s Law of Union states (A U B)’ = A’∩ B’.

The complement of two sets intersecting is equal to the complement of two sets intersecting and their union. De Morgan’s Law of Intersection states that (A∩ B)’ = A’ U B’.

Looking at the example above, we know that:

U={3,4,5,6,7}, A={6,7} and B={3,4}

Therefore, Morgan’s Law of Union – (A U B)= {3,4,6,7},

(A U B)’={5} and thus, A’∩ B’ ={5} as A’={3,4,5} and B’={5,6,7,}

So, (A U B)’ = A’∩ B’={5}

Morgan’s Law of Intersection – (A∩B)= ∅(null), (A∩B)’ ={3,4,5,6,7}

And thus, A’ U B’= {3,4,5,6,7}

As A’ = {3,4,5} and B’={ 5,6,7}

So, (A∩B)’= A’ U B’

Conclusion

When two or more sets merge in some defined way to generate a new set, it is referred to as an operation of a set. As a result, we can join sets in a variety of ways to create new ones. Any operation necessitates the use of specialised equipment and techniques, as well as problem-solving abilities. Apart from union and intersection, determining the Complement of the Set is an important approach in the field of sepsis. The complement of a set A, generally indicated by Ac (or A′), is made up of the items that aren’t in A. The absolute complement of A is the set of elements in U that are not in A when all sets under discussion are regarded to be subsets of a particular set U.