Set Operation

A collection of things is what we mean when we talk about a set. The term “Element” refers to each individual thing that is contained within a set. There are three different ways that a set can be represented. There is the statement form, the roster form, and the notation for the set builder. The term “set operation” refers to any action that is carried out on two or more sets in order to establish a connection between those sets. There are four primary categories of set operations, which can be broken down into the following categories.

• Union of sets
• Intersection of sets
• Complement of a set
• Difference between sets/Relative Complement

Let’s go through each of these processes one at a time.

Union of Sets

For any two sets A and B, the set denoted by the notation A∪B (read as A union B) is the collection of unique elements that can belong to either set A or set B or both. The number of elements in A∪B can be found by using the formula.

 n(A∪B) = n(A) + n(B) − n(A∩B), 

where the number of items in set X is denoted by the variable n(X).

In General, 

A ∪ B = {x: x ∈ A or x ∈ B}

Let us explore an example in order to gain a better comprehension of this set operation known as the union of sets: 

If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then A ∪ B is given by

 A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

Intersection of Sets

In the event when two sets, A and B, are presented, the subset of the universal set U that is formed by the intersection of A and B is the set that contains items that are shared by both A and B. The symbol for it is ” A∩B “. The following elements illustrate this operation:     

A∩B = {x : x ∈ A and x ∈ B}

Where x represents the component that is shared by both sets A and B. 

The intersection of sets A and B, can also be understood to mean the following:

      A∩B = n(A) + n(B) – n(A∪B)

Where,

n(A) represents the cardinal number of the set A,

cardinal number of set B equals n(B), where B is the set.

The cardinal number of the union of set A and set B is equal to n(A∪B).

Take this example: Let A = {1,2,3} and B = {3,4,5}

Then, A∩B = {3}; because 3 is common to both the sets.

Difference of Sets

If there are two sets, A and B, then the difference between them is equal to the set that contains items that are present in A but not in B. If there are two sets, A and B, then the difference between them is equal to that set. It is denoted by the symbols A-B.

In General,

A−B=A∩B’

Let’s say that A and B are two different sets that are made up of different elements. Let’s imagine we want 

 A = {5, 6, 8, 9, 0} 

B = { 9, 6, 0, 7, 3}

As a result,

 A – B =  {5, 6, 8, 9, 0} – { 9, 6, 0, 7, 3}    = {5,8}

It is clear to observe that when two sets A and B are subtracted from one another, the resulting set contains components that are included in set A but are absent from set B.

Similarly, B – A = { 9, 6, 0, 7, 3} – {5, 6, 8, 9, 0}   = {7, 3}

Again, the result of applying the operation B – A is the set of elements that are included in set B but are absent from set A.

Complement of Set

The set containing all of the elements in the provided universal set (U) that are not present in the given set A is referred to as the complement of the set A, and it is symbolised by the symbols A′.  

Let us consider an example:-

In the event that U = [1, 2, 3, 4, 5, 6, 7, 8, 9] and A = [1, 2, 3, 4], then…

If this is the case, the complement of the set A is denoted by the expression 

A’ = {5, 6, 7, 8, 9}.

Conclusion 

The term “set operations” refers to a group of operations that, when performed on two or more sets, result in the creation of a single set by the combining of the original sets. The set operations can be broken down into three distinct categories: the union of sets (U), the intersection of sets (⋂), and the difference between sets (-).