Venn diagram

A Venn diagram is a visual representation of sets, which are collections of distinct objects or elements. The diagram consists of overlapping circles or ellipses, each representing a different set. The overlapping regions demonstrate the intersection of sets, while the non-overlapping regions display the unique elements present in each set.

Elements of a Venn diagram

To understand Venn diagrams better, let's break down their components:

• Circles or ellipses: Each circle or ellipse represents a set. The size of the circle corresponds to the cardinality (number of elements) of that set.

• Overlap: Overlapping regions represent elements that belong to both sets.

• Non-overlapping regions: These areas represent elements unique to each set, not shared with any other set.

• Universal set: The universal set contains all elements of other sets. Elements outside the universal set are not considered for the given scenario.

Venn diagram of 3 sets

Venn diagram for three sets $A$, $B$, and $C$ is shown below:

Formulas

Some basic formulas for Venn diagrams of two and three sets:

Where $n( A)$ = number of elements in set $A$. 

Venn diagram of set operations

Union

The union of sets $A$ and $B$ comprises all elements present in $A$, $B$, or both. It is represented by the entire area covered by the two circles.

Intersection

The intersection of sets $A$ and $B$ consists of elements shared by both sets. It is depicted by the overlapping region of the two circles.

Complement

The complement of set $A$ contains all elements outside of $A$ with respect to the universal set. It is represented by the areas outside the circle of set $A$.

Difference

The difference of two sets $A$ and $B$ includes all the elements that are present in set $A$ but not in set $B$. It is represented by the area inside the circle for $A$ but outside the circle for $B$.

Real-world applications

• Survey analysis: Visualizing data from multiple-choice questions and identifying common trends or preferences among respondents.

• Genetics: Analyzing genetic traits and identifying common genetic variations among different populations.

• Market research: Segmenting consumers based on shared characteristics and preferences to target specific audiences effectively.

• Probability: Understanding the likelihood of events occurring in different scenarios by analyzing set relationships.

Conclusion

Set operations and Venn diagrams are foundational concepts in mathematics and have extensive practical applications in various fields. With their clear visual representations, Venn diagrams enable us to easily grasp complex set relationships.