2.3. Venn Diagrams

Learning Objectives

A. Use Venn diagrams to visualize relationships

between sets.

B. Use Venn diagrams to verify the equality of

sets.

C. Solve problems using Venn diagrams.

Venn Diagrams

The ideas of sets, subsets, and the operations used to combine sets can be illustrated graphically by the use of diagrams called Venn diagrams, after John Venn(1834–1923), the English mathematician and logician who invented them.

In these diagrams, we represent the universal set by a rectangle, and we use regions enclosed by simple curves (usually circles) drawn inside the rectangle to represent the sets being considered.

Venn Diagrams

For example, if A is a subset of a universal set , we can represent this universal set by the set of points in the interior of the rectangle. The interior of the circle represents the set of points in A, whereas the set of points inside the rectangle and outside the circle represents the set

Venn Diagrams

Venn Diagrams are the visualization of relation of sets. By using Venn diagrams, we can show that the commutative, associative, and distributive laws, familiar from arithmetic, apply to sets.

Verify by using Venn Diagrams!

Drawing Venn Diagrams (Intersection)

Example Given two sets A and B which are not identical and have common elements, draw a Venn diagram to represent the region corresponding to A B.

Solution

Proceed as follows:

1. As usual, the points inside the rectangle represent and the points inside the two circles represent A and B. Note that A and B overlap to allow for the possibility that A and B have points in common.

Solution

2. Shade the set A using vertical lines.

Solution

3. Shade the set B using horizontal lines. The region in which the lines intersect (cross-hatched in diagram) corresponds to A B.

Example 2 Venn Diagrams for Complements and

Intersections Draw a Venn diagram to represent the set

Solution

Drawing Venn Diagrams (Union)

The operation may be thought of as a command to shade the regions representing the sets to the left and right of the symbol with the same type lines (horizontal or vertical).

Thus, in finding A B, we shade A with, say, horizontal lines and shade B in the same way.

The union of A and B will be represented by the entire shaded region.

Example

A rectangle and two circles represent the sets , A, and B, respectively. Let’s number the four regions into which the universal set is divided.

Example(Continued)

By referring to the figure, identify the various sets with the region(s): a. A B b. 𝐴′ B c. A 𝐵′ d. A B e. 𝐴′ f. 𝐵′

Solution

a. A B is the set of elements common to A and B—represented by region 3.

b. 𝐴′ B is the set of elements not in A and in B—represented by region 2. c. A B′ is the set of elements in A and not in B—represented by region 1. d. A B is the set of elements in A or in B or in both A and B—represented by regions 1, 2, and 3. e. 𝐴′ is the set of elements that are not in A— represented by regions 2 and 4. f. 𝐵′ is the set of elements that are not in B— represented by regions 1 and 4.

Applications

Human blood is grouped according to the presence of three antigens, A, B, and Rh. Suppose that we want to use a Venn diagram to visualize all the different blood groups. We begin with three circles representing the sets of persons having antigens A, B, and Rh, respectively. As we can see, there are eight different regions, so there are eight different blood groups.

Applications( Continued )

Blood groups inside the set Rh will carry a plus sign (+) , and those outside Rh will carry a minus sign (–) .

Blood with neither the A nor the B antigen will be labelled type O.

Example 4 Identifying Blood Groups

Draw a Venn diagram to identify all possible blood groups.

Solution

The eight possible groupings of blood are A–, A+, B–, B+, AB–, AB+, O–, and O+, where A+ means that the person has both A and Rh antigens, A– means that the person has antigen A but not Rh, and similarly for the remaining groupings.