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Union and Intersection Help

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Union of sets is denoted by ∪ while intersection is denoted ∩

To easily understand the meaning of union and intersection it is important to first define a set.

A set can be defined as a collection of objects or items that is well-defined. Members of a set are referred to as elements. Sets are said to be either equal or unequal. Two or more sets are equal if they have the same elements in them. Conversely, they are unequal if there are no common elements. A set with no elements is referred to as an empty or null set. Assuming that A and B are sets, their union is a new set which contains all the elements in the two sets. The union of sets is usually denoted by and will be written as A∪B. Intersection of two sets on the other hand refers to the common elements between two or more sets. The intersection of sets is usually denoted by . Therefore, the intersection of A and B will be written as AB.

A useful way to remember the symbols is ∪nion and i∩tersection

Union and Intersection Examples using Venn diagrams

A Venn diagram is a diagram that shows all the possible and logical relations and difference between a finite collection of two or more sets using circles. In a Venn diagram, elements and intervals are depicted as points in the plane while sets are regions inside the circles. In other words, each circle represents a given set and the members are indicated inside the circle. An example of a Venn diagram illustration of sets and formula is as shown below;

Example 1:

If A = {1, 3, 5, 7, 9) and B= {4, 5, 7, 9). The Union and intersection of sets A and B as would be given by a calculator are

A ∪ B = {1, 3, 4, 5, 7, 9}

A B = {5, 7, 9}

This can be illustrated by Venn diagram as shown below

Union and Intersection example

Union Example

A∪B is any region including either A or B

union example

Intersection Example

AB is any region including both A and B as shown below

Union and Intersection Example 2

If X = {a, e, i, o, u) and Y= {a, i, u). The Union and intersection of sets A and B will be

X ∪ Y = {a, e, i, o, u}

X Y = {a, i, u}

Therefore set Y can be referred to as a subset of X since every element in set Y is in Set B.

Laws Associated with Union and Intersection

There are 3 main types of laws that are associated with the union and intersection of sets. They include: commutative laws, associative laws and distributive laws

Commutative Laws

The commutative law states that for any sets X and Y,

X ∪ Y=Y ∪ X

X Y = Y X

Associative Laws

The commutative law states that for any sets X, Y and Z, we have

(X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)

(X Y) Z = X (Y Z)

Distributive Law

The distributive law states that for any sets X, Y and Z, we have

X (Y ∪ Z) = (X Y) ∪ (X Z)

X ∪ (Y Z) = (X ∪ Y) (X ∪ Z)

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