**Venn Diagram**s are used to display sets. A Venn diagram consists of one or more circles within a rectangle.

The rectangle defines the Universal Set, and each circle within the rectangle is a different set. Elements not in any set are shown within the rectangle but not inside a circle.

For example, if two dice are thrown then ε = {2,3,4,5,6,7,8,9,10,11,12). Set P consists of Prime Numbers and set Q consists of odd numbers:

Venn Diagrams can be used to determine the interaction of sets. The areas of white in the diagrams below show the result of the operation:

`P nn Q = {3, 5, 7, 11}` and `P uu Q = {2, 3, 5, 7, 9, 11}`.

Venn diagrams can also be used to show totals, rather than individual items.

Given a set of integers from 1 to 15, set A = {multiples of two} and set B = {multiples of three}. The Venn Diagram has been started. Show the elements of A ∩ B.

`A={2,4,6,8,10,12,14}` and `B={3,6,9,12,15}`

Part *x* is A overlapping B, which is `A nn B = {6, 12}`.

Answer: `A nn B = {6, 12}`

A field trip consists of 30 students. 8 of these students do not study a language.

21 students study French. 18 students study German. Complete the Venn Diagram, finding values for *w*, *x*, *y* and *z*.

The 8 students that do not form part of any set are shown outside the sets in position *w*.

The total for French and German is 21 + 18 = 39, but there are only 22 students that study a language. The number of students studying two languages is 39 - 22 = 17 = *y*.

There are 21 students studying French less the 17 studying both = 4 = *x*.

Similarly for German: 18 - 17 = 1 = *z*.

Check it: *w* + *x* + *y* + *z* = 30✔

Answer: *w* = 8, *x* = 4, *y* = 17, *z* = 1

See also Highest Common Factor and Lowest Common Multiple

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