Complement of an intersection

The complement of the set X ∩ Y is the set of elements that are members of the universal set U but not members of X ∩ Y. It is denoted by (X ∩ Y) ’.

The symmetric difference of two sets is the collection of elements which are members of either set but not both - in other words, the union of the sets excluding their intersection. Forming the symmetric difference of two sets is simple, but forming the symmetric difference of three sets is a bit trickier.


Suppose U = set of positive integer…

Union of Sets

The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’

The following table gives some properties of Union of Sets: Commutative, Associative, Identity and Distributive. Scroll down the page for more examples.
Properties of Union of Sets

Example :

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.


X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} …

Intersection of sets

The intersection of two sets X and Y is the set of elements that are common to both set X and set Y. It is denoted by X ∩ Y and is read ‘X intersection Y ’.


Draw a Venn diagram to represent the relationship between the sets

X = {1, 2, 5, 6, 7, 9, 10} and Y = {1, 3, 4, 5, 6, 8, 10}


We find that X ∩ Y = {1, 5, 6, 10} ← in both X and Y

For the Venn diagram,

Step 1 : Draw two overlapping circles to represent the two sets.

Step 2 : Write down the elements in the intersection.

Step 3 : Wri…

Venn Diagram

A Venn Diagram is a pictorial representation of the relationships between sets.

We can represent sets using Venn diagrams. In a Venn diagram, the sets are represented by shapes; usually circles or ovals. The elements of a set are labeled within the circle.

The following diagrams show the set operations and Venn Diagrams for Complement of a Set, Disjoint Sets, Subsets, Intersection and Union of Sets. Scroll down the page for more examples and solutions.
Set Operations and Venn Diagrams

The set of al…


The cardinality | S | of set is the number of members in a set.

1. Set A = { apple, atis, avocado} contains 3 elements, therefore, A has a cardinality of 3.

|A|= 3

2. Set C = { }, |C|= 0