Venn diagrams

John Venn was an English mathematician who used diagrams to illustrate the relationships among sets. These diagrams help to organise information about probabilities, which can be useful when solving conditional probability problems.

Suppose that U=\left\{a, b, c, d, e, f, g, h, i, j\right\}

A=\left\{a, c, h, j\right\}

B=\left\{c, h, g, i, j\right\}

The diagram above shows that: the set A contains a, c, h, j and the set B contains the elements c, h, g, i, and j. It also shows that the universal set U contains a, b, c, d, e, f, h, i, j.

The elements c, h, and j are found at the intersection of A and B, and we denote this by:

A \cap B= \left\{c, h, j \right\}

Together, A and B contain \left\{a, c, h, j, g, i \right\}, which we denote by:

A \cup B= \left\{a, c, h, h, i, j, g \right\}

  • \cap is called the intersection
  • \cup is called the union of A and B

The intersection \cap of sets A and B, written as A \cap B, which is the set that includes all the elements that are common to both A and B.

The union \cup, of sets A and B, written as A \cup B, which is the set containing all the elements of both sets A and B.

A = \left\{1, 3, 5, 7, 9, 11, 13, 15 \right\}

B = \left\{2, 3, 7, 7, 11, 13 \right\}


A \cap B= \left\{3, 7, 11, 13 \right\}

A \cup B= \left\{1, 2, 3, 5, 7, 9, 11, 13, 15 \right\}


The image below is a Venn diagram, which shows the devices owned by the people surveyed.

What does each number mean? And how many people were surveyed?

The information that is put into this diagram:

  • 120 people were surveyed
  • 52 owned just a laptop
  • 45 owned just a tablet
  • 23 owned both a laptop and a tablet