Union, Intersection, and Complement of Sets

Introduction

Union, intersection, and complement of sets are some of the set operations, which are used in the ‘set theory’ of mathematics that studies sets. You can describe a set as a group of any mathematical or non-mathematical elements (or members), multiple sets carry out operations with each other to solve math problems. Its operations are unions, intersections, and complements. We are going to discuss set operations in detail.

About Sets and Set operations

A set is the group of any particular elements like numbers, symbols, variables, etc. The set theory includes many different types of sets, and many operations take place to interpret their results. Here are some of the fundamental set operations of set theory. Union of sets, the intersection of sets, and complement of sets, all three set operations are important for set-related problems.

Union Sets

An operation in which two sets are combined can be termed as the union of sets. Union operation is denoted by ‘U’ in between. [ Note: P and Q are observational denotes taken throughout the article]. The general formula of union set: P∪Q = {x: x ∈ P or x ∈ Q}

Properties

Property	Description	Equation
Commutative	A commutative property describes the sets that are commutative and can be presented in any order	P ∪ Q = Q ∪ P
Associative	Parenthesis can be rearranged as the result will be the same	(P ∪ Q) ∪ R = P ∪ (Q ∪ R)
Idempotent	The same elements in sets after union give the same set as the result	P ∪ P = P
Property of ∅ or Identity Law	Set union with any empty set has no changes in the result	P ∪ ∅ = P
Universal Set	Universal set contains all elements, as the same elements don’t repeat, the universal set has no change in any union	A ∪ U = U

Examples

E.g. Let’s suppose, sets P & Q are performing union operations. The equation would be P ∪ Q.   P = {1, 3, 5, 6}; Q= {1, 2, 6, 7}

• P U Q = {1, 2, 3, 5, 6, 7}

E.g.  S = {every natural odd number till 10}; T = { every natural even integer till 10}. Describe       S ∪ T.

• S = {1,3,5,7,9}; T = {2,4,6,8,10}
• S U T = {1,2,3,4,5,6,7,8,9,10}

These were the examples of sets performing union operation in them, the result we get is the combination of all elements from both sets. The result of the operation doesn’t include repeated elements and hence does not increase their cardinality.

Intersection sets

The intersection of sets is the operation that interprets the common elements from both sets that operated as a result. Intersection sets are denoted by ‘∩’ in set operations between two sets. The general formula of intersection sets: P∩Q = {x: x ∈ P and x ∈ Q}

Properties

Property	Description	Equation
Commutative	A commutative property describes the sets that are commutative and can be presented in any order	P ∩ Q = Q P
Associative	Parenthesis can be rearranged as the result will be the same	(P ∩ Q) ∩ R = P ∩ (Q  ∩ R)
Distributive	Intersection gets upon union and Union upon the intersection	(P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R); (P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)
φ & ∪ (phi and union) property	When an intersection occurs with an empty set there is no common element	∩ P = ; U ∩ P= P

Examples E.g. Let’s suppose, sets P & Q are performing Intersection. The equation would be PQ. P = {1, 3, 5, 6}; Q= {1, 2, 6, 7}

• P Q = {1,6}

E.g.  S = {every natural odd number till 10}; T = { every natural even integer till 10}. Describe ST.

• S = {1,3,5,7,9}; T = {2,4,6,8,10}
• S T = {𝝋}

In the above-performed set operations, we can conclude that the common elements present in both sets are the result of intersections. Almost the same examples are taken for intersection sets as of union sets to show the difference between them.

Complement Sets

The complement of sets is a different operation from intersection and union sets. In complement set operations, the set is filled with the elements of a given universal set excluding elements present in the other set in complement of intersection set and it is reversed in complement of union sets. The complement of the set is denoted by ‘Ac’. The general formula of the complement of a set is P′ = {p : p ∈ U and p ∉ P’}

Properties

The complement of the union of sets results in universal sets as all elements absent in set P gets filled by P’ which is a complement of set P. P ∪ P’ = U.	Completion of the intersection of sets results in an empty set as all elements present in set P gets excluded which are present in P’. P ∩ P’ = φ	Double complement law:  if the complement set is executed with a complemented set of P’ of it, the result will be the set P.
Empty set law: When a universal set is executed with its complement set, this operation will result in an empty set, And reverse of this for the empty set.	De Morgan’s laws: 1.) (P ∪ Q)’ = P’ ∩ Q’ 2.) (P ∩ Q)’ = P’ ∪ Q’.	These de Morgan’s laws can be applied to any finite set.
Examples	U = {1, 3, 4, 5, 7, 8, 10}, P = {4, 8, 10}.	P’ = U – P = {1, 3, 4, 5, 7, 8, 10} – {4, 8, 10} P’ = {1, 3, 5, 7}

Conclusion

Above from here, we came through the union of sets, the intersection of sets, and the complement of sets. We learned their definitions, formulas, properties, and multiple examples also.