Venn Diagrams, Regions & Numbers in Regions

KEY

Venn diagrams are often used to represent sets of objects, numbers, or things. A Venn diagram consists of a universal set U represented by a rectangle. Sets within the universal set are usually represented by circles.

/ U = {2, 3, 5, 7, 8}
A = (2, 7, 8}
A’ = {3, 5}

Intersection

consists of all elements common to both A and B. It is the shaded region where the circles representing A and B overlap. /

Union

consists of all elements in A or B or both. It is the shaded region which includes both circles. /

Subsets

If , then every element of B is also in A. The circle representing B is placed within the circle representing A. /

Disjoint or mutually exclusive

Disjoint sets do not have common elements. They are represented by non-overlapping circles. /

Example. SupposeU = {1, 2, 3, 4, 5, 6, 7, 8}. Illustrate on a Venn diagram the sets:

  1. A = {1, 3, 6, 8} and B = {2, 3, 4, 5, 8}
/
  1. A= {1, 3, 6, 7, 8} and B = {3, 6, 8}

Example. SupposeU = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Illustrate on a Venn diagram the sets A = {2, 4, 8} and
B = {1, 3, 5, 9}.

Shading

We can use shading to show various sets on a Venn diagram.

Set / / / /
Shaded region: / A / / B’ /

Example. Shade the following regions for two intersecting sets A and B:

Numbers in regions

There are many situations where we are only interested in the number of elements of U that are in each region. We do not need to show all the elements of the diagram, so instead we write the number of elements in each region in parenthesis.

Example. How many elements are there in:

/
  1. P
n(P) = 7 + 3 = 10 /
  1. P, but not Q
n(P, but not Q) = 7
  1. Q’
n(Q’) = 7 + 4 = 11 /
  1. Q, but not P
n(Q, but not P) = 11
n() = 7 + 3 + 11 = 21 /
  1. Neither P nor Q?
n(neither P nor Q) = 4

Example. Given n(U) = 30, n(A) = 14, n(B) = 17, and n() = 6, find:

  1. n()
n() = 25 /
  1. n(A, but not B)
n(A, but not B) = 8