Unions, Intersections, and Complements

Union, intersection, and compliments of any particular set are essential parts of set theory and operation topics. To know the compliment detail of these terms. It is crucial to understand the basics of the set theory of mathematics. Set is a compilation and collection of observations, entities, or elements. Every set has entity observations associated with a specific type or medium. This set is known as the Universal set. It is represented by U.

 Operations of the set are the mathematical functions that are performed between any two sets or more sets to obtain a specific solution. In the chapter on set theory, different operations are performed on a given particular type of set, but three prominent set operations are there:-

1. Complement of sets

2. The intersection of sets (∩)

3. Union of sets (∪

Every set A in this set theory topic is always a subset of given Universal set U. The compliment of any set B has all the terms or members, attribute from the Universal which is not part of original set B. Compliment set is always is denoted by B’.

Different operations of set

 The intersection of the set (∩)

Suppose we have two general sets X and Y. The intersection(∩) of X and Y is the subset or portion of the universal set represented by symbol U, which consists of the common components to both the set X and Y. Intersection of sets is depicted by the symbol ‘∩’. This is technique is denoted  by:

X ∩Y = {z : z ∈ X and x ∈ Y}

Where z is the common element of both sets X and Y.

The intersection of sets A and B, can also be inferred as:

X∩Y = n(X) + n(Y) – n(X∪Y)

Where,

n(X) = leading number of set X,

n(Y) = leading number of set Y,

n(X∪Y) = leading number of union of both set X and Y.

Specimen: Let X = {a,b,c} and Y = {c,d,f}

Then, X∩Y = {c}; because c is common to both the sets.

Union of sets (∪)

Suppose we have two given sets X and Y, then the union of X and Y sets is always equal to the set that has all the components, available in set X and set Y. This operation can be expressed as:

X ∪ Y = {z: z ∈ X or x ∈ Y}

Where x is the components present in both sets X  and Y.

For instance: If set X = {a,b,c,d} and Y= {d,e}

Then, Union of sets, X ∪ Y = {a,b,c,d,e}

Complement of sets

For instance, U is a universal set and P is any given subset of U then the complement of P  is the set of all the components of the universal set U excluding all components of P.

P′ = {z : z ∈ U and a ∉ P}

Venn diagram

A Venn diagram is a graphical representation of a given set.  It showcases the relation among sets, its operation in a pictorial manner. It is formed using the crossing, bisecting, overline circles. Venn diagram is also known by the name logic diagram, set diagram.

It is the best approach to demonstrate various operations of the set for example compliment, difference, union, and the intersection of a given set.

Venn diagram of set operations

In the set theory, the operations performed on sets are:

• Complement of set

• Intersection of set

• Union of set

Different properties of sets operations

Compliment of set properties

• φ′ = U

• (A ∪ B)′ = A′ ∩ B′

• U′ = φ

• A ∪ A′ = U

• (A ∩ B)′ = A′ ∪ B′

• A ∩ A′ = φ

The intersection of two sets 

A intersection B is given by A∩ B = {x : x ∈ A and x ∈ B}.

Intersection of set properties

• A ∩ B = B ∩ A

• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

• A ∩ A = A

• φ ∩ A = φ

• (A ∩ B) ∩ C = A ∩ (B ∩ C)

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

•  U ∩ A = A

Union of two sets

A Union B is given by: A ∪ B = {x | x ∈A or x ∈B}.

Union operation properties

• A ∪ A = A

• A ∪ φ = A

• A ∪ B = B ∪ A

• U ∪ A = U

• (A ∪ B) ∪ C = A ∪ (B ∪ C)

Conclusion

In the stream of mathematical studies, Union, intersection, and complement of sets play an important role. This relation and interrelation. We learned about various set operations, including union, intersection, and complement in this article. Also, their different associated properties. Venn diagram and its relation with set theory. In accordance with the mathematical studies, the Venn diagram represents different operations associated with set theory. These operations include complement, union, and intersection.