1C  Venn Diagrams

It is easier to think about something if you have a picture of it in front of you; perhaps a visual stimulus helps to shift the brain into gear. That is why, when working on a math problem, probably the first thing you should do is draw a picture. Indeed, just carrying out the mechanics of making a sketch forces one to stop daydreaming and to start thinking seriously about the problem. The Englishman John Venn (1834 - 1923) introduced the systematic use of certain sketches, which we now call Venn diagrams, as visual aids in the study of sets and logical reasoning.

Below are Venn diagrams for two sets A and B. We imagine that A consists of all points inside the circle labeled A, and we make the analogous assumption for B. The large rectangle containing both circles represents the universe U. The diagram on the left shows A and B intersecting, while on the right A and B are disjoint. (Usually circles are used to represent sets, but also other shapes, such as ovals or rectangles, may be used. The use of colors is unnecessary, but it makes the pictures more attractive.)  Intersecting Sets Disjoint Sets

Venn diagrams can illustrate other situations. Below are diagrams for A Ì B, B Ì A, and A = B:   A Ì B B Ì A A = B

We can draw attention to certain regions by shading, cross-hatching or coloring them. Following, from left to right, we shade ~A, A ∩ B, and A È B:   Shaded : ~A Shaded : A ∩ B Shaded : A È B

In the left sketch, ~A contains those points in the universe rectangle U but not in the circle A. In the middle sketch, A ∩ B contains the points common to circles A and B. On the right, A È B consists of all points in either circle A or circle B (or perhaps in both circles).

We can use Venn diagrams also to verify set identities. The De Morgan laws (presented by Augustus De Morgan of England, 1806 - 1871), are the formulas

~(A È B) = ~A ∩ ~B           ,           ~(A ∩ B) = ~A È ~B .

The left formula states that those elements not in either A or B are exactly those elements both not in A and not in B. The right formula asserts that those elements not in both A and B are those elements either not in A or not in B. (These statements should make sense if you mull them over awhile.) We will use Venn diagrams to justify the De Morgan law on the left, leaving the analogous verification of the law on the right as an exercise. First we sketch general intersecting sets A and B. In the first line of diagrams below we shade the set A È B and then its complement ~(A È B). In the second line we shade ~A, then ~B, and then the common points of these regions to get ~A ∩ ~B. We find that the final shadings are the same in the two lines, thereby confirming the identity.  A È B ~(A È B)   ~A ~B ~A ∩ ~B At right is a Venn diagram for three intersecting sets, A, B, and C. Note that the three circles divide the universe rectangle U into eight distinct regions. The center brown region is A ∩ B ∩ C; it contains those points common to all three sets. The union A È B È C, composed of all points in at least one of the three sets, envelops all but the gray region. Also, A ∩ B encloses the green and brown regions, and A È B contains all but the gray and blue regions. Other regions can be a little more complicated to describe. The green region is A ∩ B ∩ ~C (those points in both A and B but not in C), while A ∩ ~B ∩ ~C is the red region.

example 1

We shade the region A ∩ (B È ~C) in a Venn diagram for three sets. These will be the points which are in the A circle, and either in the B circle or not in the C circle. You might be able to figure this one out in your head by staring at the picture for awhile, but there is a systematic method which works for even more complicated problems. First we draw the Venn diagram and label the eight distinct regions from 1 to 8. Then, as demonstrated below, we determine which numbered regions make up the expression we want to shade. Lastly, we do the shading. A : 1, 2, 4, 5 B : 2, 3, 5, 6 C : 4, 5, 6, 7 ~C : 1, 2, 3, 8 B È ~C : 1, 2, 3, 5, 6, 8 A ∩ (B È ~C) : 1, 2, 5

EXERCISES 1C

1. In a Venn diagram for two intersecting sets A and B, shade
1. A ∩ ~B,
2. ~A È B.

2. Use Venn diagrams to verify the De Morgan law ~(A ∩ B) = ~A È ~B.

3. Use Venn diagrams to verify that (A ∩ B) È (A ∩ ~B) = A.

4. Draw a Venn diagram for three sets A, B, and C, illustrating that
1. A Ì B Ì C,
2. A and B are disjoint, but both are subsets of C,
3. A and B intersect, neither is a subset of the other, but both are subsets of C,
4. A Ì B, A Ì C, and neither B nor C is a subset of the other.

5. In a Venn diagram for three general intersecting sets A, B, and C, shade
1. ~A ∩ B ∩ C,
2. ~A ∩ B ∩ ~C,
3. (B È C) ∩ ~A,
4. A È (~B ∩ C),
5. ~(A ∩ B) ∩ C,
6. ~C ∩ (~B È A).