Book of Proof - Amazon S3

Venn Diagrams 23
Compare the diagrams for (A ∪B)∩C and A ∪(B ∩C) in Figures 1.9 and
1.10. The fact that the diagrams are different indicates that (A ∪ B) ∩ C ≠
A ∪ (B ∩ C) in general. Thus an expression such as A ∪ B ∩ C is absolutely
meaningless because we can’t tell whether it means (A∪B)∩C or A∪(B∩C).
In summary, Venn diagrams have helped us understand the following.
Important Points:
• If an expression involving sets uses only ∪, then parentheses are optional.
• If an expression involving sets uses only ∩, then parentheses are optional.
• If an expression uses both ∪ and ∩, then parentheses are essential.
In the next section we will study types of expressions that use only ∪
or only ∩. These expressions will not require the use of parentheses.
Exercises for Section 1.7
1. Draw a Venn diagram for A.
2. Draw a Venn diagram for B − A.
3. Draw a Venn diagram for (A − B) ∩ C.
4. Draw a Venn diagram for (A ∪ B) − C.
5. Draw Venn diagrams for A∪(B∩C) and (A∪B)∩(A∪C). Based on your drawings,
do you think A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)?
6. Draw Venn diagrams for A∩(B∪C) and (A∩B)∪(A∩C). Based on your drawings,
do you think A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)?
7. Suppose sets A and B are in a universal set U. Draw Venn diagrams for A ∩ B
and A ∪ B. Based on your drawings, do you think it’s true that A ∩ B = A ∪ B?
8. Suppose sets A and B are in a universal set U. Draw Venn diagrams for A ∪ B
and A ∩ B. Based on your drawings, do you think it’s true that A ∪ B = A ∩ B?
9. Draw a Venn diagram for (A ∩ B) − C.
10. Draw a Venn diagram for (A − B) ∪ C.
Following are Venn diagrams for expressions involving sets A,B and C. Write the
corresponding expression.
C
C
C
C
11. A B
12. A B
13. A B
14. A B