# 3 Properties of set operations with examples

The properties of set operations are similar to basic mathematics operations. Sets are defined as a collection of well-defined objects of the same kind. In this article, we will be looking at set operations and the properties of set operations.

## set operations

There are three main sets of operations in mathematics. This includes the Intersection of sets (∩ ), the union of sets (U), and the complement of a set A’. Let’s look at this set’s operations one after the other.

Intersection of sets: two or more sets are said to have an intersection when elements or members of one set are found in the elements or members of the other set. For example, if set A ={2, 3, 4, 5} and set B ={6, 5, 4, 7}, the intersection of A and B i.e A ∩ B = {4,5}.

Union set: is formed by putting members of one set together with members of another set. For example, if set Q={3,4,5} and set P ={1,2,3,4}, the union of Q and P i.e Q U P={1,2,3,4,5}. Note that members in a union set are written from lowest to highest and any member that belongs to both sets is written once.

Complement of a set: a set of objects that belong to the universal set but are not found in the given set is known as a complement of a set. For example, if U is the universal set {1,2,3,4,5,6,7} and set B={3,5,7} is a subset of the universal set, then the complement of set B. i.e B’ ={1,2,4,6}.

## Properties of set operations

The properties of set operations are similar to basic operations in mathematics. There are three properties of set operations which includes commutative property, associative property, and distributive property.

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Commutative property: the commutative property of sets is defined as A ∩ B = B ∩ A and A U B = B U A. This means that the Intersection of sets and Union of sets are commutative. For example, if A={3,4,5,6,7} and B={5,7,9,11}, then A ∩ B=B ∩ A ={5,7}. Also, A U B = B U A ={3,4,5,6,7,9,11}.

Associative property of sets: The associative property of sets is defined as (A ∩ B)∩C =A ∩(B ∩ C) and (A U B) U C. Therefore, the intersection of sets and Union of sets are associative. For example, if A ={2,4,6}, B={2,3,5,7} and C={1,2,3,4,5} then (A ∩ B)∩C = A ∩(B∩C)= {2} and ( A U B) U C=AU( B U C)={1,2,3,4,5,6,7}.

Distributive property of sets: The distributive property of sets is defined as AU(B∩C)=(AUB)∩(AUC). This shows the union of sets is distributive over the Intersection of sets. Also A ∩(BUC)=(A ∩B)U(A ∩ C) and A∩(BUC)=(A ∩B)U(A ∩C), A ∩(BUC)=(A ∩B)U(A ∩C), then the Intersection of sets is distributive over the union of sets. Let’s look at some examples, if A={2,3,4,5}, B={2,3,5,7} and C={2,4,6,8}, then AU(B ∩ C)=(AUB)∩(AUC) ={2,3,4,5}. Also A∩(BUC)=(A ∩B)U(A ∩ C) ={2,3,4,5}.