Set Theory and Venn Diagrams

A set is a well-defined collection of objects. Each member of a set is called an element. When listing the elements contained within a set, set brackets, { }, are used to enclose the elements belonging to a set. For example, the set of prime numbers from
1 to 20, is written as {2, 3, 5, 7, 11, 13, 17, 19}.

Relationships Among Sets

The intersection of two sets A and B, written as , consists of those elements
that are common to both set A and set B.
For example, if A = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} and B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20},
then = {6, 12, 18}

The union of two sets A and B, written as , consists of those elements
that are common to set A or set B or both.
                          For example, if A = {cat, dog, hamster, goldfish} and B = {rat, mouse, beaver, hamster},
              then = {cat, dog, hamster, golfish, rat, mouse, beaver}

Venn Diagrams

One useful way of understanding the relations between sets is by using Venn diagrams.

John Venn, pictured at the right, popularized the idea of Venn Diagrams. He lived from 1834 to 1923 in England. He was a priest and taught at Gonville and Caius College of Cambridge.

  

A typical Venn diagram uses overlapping circles to represent groups of items or ideas that share common properties. In a venn diagram, all elements of a set are contained within a given circle and elements which are shared between two sets are contained within the overlapping regions of the circles.

A venn diagram which illustrates the intersection and union of two events A and B is shown at the right. The rectanglular region represents the universal set, U, which is the set that contains all elements being discussed.

The site, http://www.mathsnet.net/asa2/modules/s13venn.html, contains several Venn diagrams; each of which illustrates a particular relationship between two sets.

 


Example
Let the universal set be all even numbers from 2 through 60.
Let A represent all numbers from 2 through 60 which are multiples of 4 .
Let B represent all numbers from 2 through 60 which are multiples of 10.
Let C represent all numbers from 2 through 60 which are multiples of 6.
Determine the elements for each of the following.
Recall that AUB means "the elements in A or B or both" and means "the elements common to A and B"