Venn diagrams

Venn Diagrams, named after the English mathematician John Venn, provide a visual representation of how different sets relate to each other.

A set is just a collection or group of things, like a list. For example, if you have a set of favourite colours, it might be {red, blue, green}. Venn Diagrams help us see how different sets, or groups of things, have items in common or are different.

Basics of Venn Diagrams

There are some terms and definitions that are important for you to remember. Let’s use the diagram below as an example.

Now let’s look at the key terms and definitions:

  • Set: A collection of distinct objects. In this case, A, B and U are sets.
  • Universal Set (U or ξ): This is a set that contains all the elements under consideration in a particular scenario. For example, if we are talking about the letters from ‘a’ to ‘j’, the universal set would be U = \lbrace a, b, c, d, e, f, g, h, i, j \rbrace. This set can be represented by a box in a Venn Diagram.
  • Intersection (\cap): This represents the elements that two or more sets have in common. For the sets, A and B provided, the intersection (the elements they both share) is A \cap B = \lbrace c, h, j \rbrace.
  • Union (\cup): This represents all the elements that are in either of the sets or both. For the sets, A and B, the union is A \cup B = \lbrace a, c, g, h, i, j \rbrace.

Drawing Venn Diagrams

The process starts by first drawing a rectangle which represents the Universal Set. Inside this, we draw circles (or sometimes ovals) to represent the other sets. Overlapping areas of circles contain common elements between those sets.

From our earlier example, sets A and B have an intersection at c, h j. This overlap or intersection is represented by the area where the two circles for A and B overlap.

Solving Probability Problems with Venn Diagrams

Venn Diagrams become especially useful when dealing with problems that have multiple events, such as those involving AND, OR, and NOT operations.

For example, consider two more sets:

A = \lbrace 1, 3, 5, 7, 9, 11, 13, 15 \rbrace

B = \lbrace 2, 3, 7, 11, 13 \rbrace

If we are asked to find out which numbers are in both set A and set B, we can determine the intersection A \cap B = \lbrace 3, 7, 11, 13 \rbrace.

If asked which numbers are in either set A or set B (or both), we find the union: A \cup B = \lbrace 1, 2, 3, 5, 7, 9, 11, 13, 15 \rbrace.

Practice Questions

Question 1:

Using the sets:

  • A = \lbrace 1, 3, 5, 7 \rbrace
  • B = \lbrace 3, 7, 9, 11 \rbrace

Find A \cap B

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The numbers 3 and 7 appear in both sets, so A B = \lbrace 3, 7 \rbrace


Question 2:

For the same sets:

A = \lbrace 1, 3, 5, 7, 9, 11, 13, 15 \rbrace

B = \lbrace 2, 3, 7, 11, 13 \rbrace

Find A \cup B

Combine all the unique numbers from both sets to get A ∪ B = \lbrace 1, 2, 3, 5, 7, 9, 11, 13, 15 \rbrace

Question 3:

In a school of 500 students:

  • 200 play football
  • 250 play basketball
  • 50 play both football and basketball

How many students do not play football or basketball?

First, find the total number of students who play at least one of the sports: 200 + 250 - 50 = 400 (subtracting 50 to avoid double counting those who play both sports). Now, subtract this from the total number of students: 500 - 400 = 100.

So, 100 students do not play football or basketball.


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