What are Venn Diagrams?

Venn Diagrams are a great way to visualize sets. Venn diagrams are simple pictures used to represent sets and their relationships. They are drawn with shapes, usually circles, to show how groups overlap or differ. Each circle stands for a set, and the shaded parts show the results of set operations like union, intersection, and complement.

Why are they called Venn diagrams?

They are named after the English logician John Venn, who introduced them in the 1880s as a way to visualize logical relations between classes. Although similar diagrams existed earlier, Venn standardized their use in mathematics and logic.

Why do we use them?

Venn diagrams are used to make abstract set operations concrete and easy to understand. They help in reasoning about inclusion, exclusion, and overlap between groups. Common uses include probability (e.g. events happening together or separately), logic (e.g. truth of combined statements), and classification (e.g. sorting data into categories).

Reading the first graph: Basic operations on two sets

$A \cap B$ — the overlap. Elements that are in both A and B.

$A \cup B$ — the combined region. Elements that are in A, or in B, or in both.

$A - B$ — only in A. Elements in A that are not in B.

$B - A$ — only in B. Elements in B that are not in A.

Tip: Click a button; only that region should be shaded.

Reading the second graph: Complements

$A^c$ — everything in the universal set U that is not in A.

$B^c$ — everything in U that is not in B.

$(A \cap B)^c$ — outside the overlap. Everything except the common part of A and B.

$(A \cup B)^c$ — outside both. Everything in U that is in neither A nor B.

Tip: Use the buttons to see how complements carve out regions of U.