Venn Diagram Basics

The Oxford Dictionary defines the Venn diagram as “a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.” The venn diagram is a special case of the Euler diagram in that it shows all of the possible relationships between different sets. The Euler diagram shows only relevant relationships.

Structure of Venn Diagrams

Venn diagrams, also known as set diagrams or logic diagrams, use overlapping circles to visually show the relationships between multiple sets of items. The circles are placed inside a rectangle that represents the universe, or universal set. Each circle in the rectangle represents all of the elements in a subset of the universal set. Everything outside a circle is the complement of that circle. For example, if a set is labeled A, everything outside the set A represents the complement of A. If a second set is labeled B, everything outside the set B represents the complement of B. The intersection of the subsets, or circles, represents all of the elements that are common to both subsets A and B.

For a Venn diagram with two circles, the universal set is identified as the intersection of the two subsets and their complements. The union of the two subsets A and B is denoted by A ∪ B. Likewise, the intersection of the two subsets A and B is denoted by A ∩ B.

Purpose of Venn Diagrams

Venn diagrams offer a convenient way to visualize the logical relationships between different sets of items, or data. They are useful in set theory, the branch of mathematical logic that studies sets, but also offer a way to visualize real-life problems and are widely used in many fields. Because Venn diagrams offer a clear way to visualize the logical relationships among different groups or categories, they are popularly used in presentations, market analysis, or reports in any area where an analysis of the similarities or dissimilarities of groups is the focus, such as in the following examples:

• Venn diagrams are used in schools to teach basic math concepts as well as in higher education to teach advanced mathematics.
• In reading comprehension courses, students use Venn diagrams to analyze the ideas they read about.
• Programmers use Venn diagrams to compare and contrast computer languages and develop hierarchies.
• Statisticians use Venn diagrams as tools to help predict likely outcomes. Different sets are analyzed to determine similarities and differences.
• Venn diagrams are in logic to prove or disprove the validity of logical arguments. For example, if all boys are mammals, and my child is a boy, then my child is a mammal.

These are just a few of the many uses in which Venn diagrams are used.

Venn diagrams offer a convenient and visual way to organize, compare and contrast particular sets of data. They are closely related to Euler diagrams but are different in that they include sets in which no items exist.