Solutions to Problem Set 16

Math 310-01/02

10-20-2017

1. Suppose that the universe is $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ , and that

$$A = \{1, 3, 5, 7, 9\} \quad\hbox{and}\quad B = \{4, 5, 6, 7, 8, 9, 10\}.$$

List the elements of the following sets.

(a) $\overline{A}$ .

(b) $A - B$ .

(c) $A \cap B$ .

(d) $A \cup B$ .

(a)

$$\overline{A} = \{2, 4, 6, 8, 10\}.\quad\halmos$$

(b)

$$A - B = \{1, 3\}.\quad\halmos$$

(c)

$$A \cap B = \{5, 7, 9\}.\quad\halmos$$

(d)

$$A \cup B = \{1, 3, 4, 5, 6, 7, 8, 9, 10\}.\quad\halmos$$


2. List all the subsets of $\{a,
   \{b, c\}\}$ .

$$\emptyset, \quad \{a\}, \quad \{\{b, c\}\}, \quad \{a, \{b, c\}\}.\quad\halmos$$


3. Draw Venn diagrams for the following sets. Shade the part of the diagram corresponding to the set.

(a) $A - (A \cap B)$ .

(b) $(A - C) \cup (C \cap B)$ .

(a)

$$\hbox{\epsfysize=1.5in \epsffile{vennc.eps}}\quad\halmos$$

(b)

$$\hbox{\epsfysize=1.5in \epsffile{vennd.eps}}\quad\halmos$$


4. Draw Venn diagrams which illustrate the following sets.

(a) $(A \cup B) - (A \cap B)$ .

(b) $(A - B) \cup (B - C)$ .

(a)

$$\hbox{\epsfysize=1.5in \epsffile{venna.eps}} \quad\halmos$$

(b)

$$\hbox{\epsfysize=1.5in \epsffile{vennb.eps}} \quad\halmos$$


5. Let A, B, and C be sets. Prove the following statement using the definitions of union, intersection, and complement and the rules of logic. Justify each step.

$$(A \cap B) \cap (A \cap C) = A \cap (B \cap C)$$

$$\matrix{x \in (A \cap B) \cap (A \cap C) & \iff & x \in (A \cap B) \land x \in (A \cap C) & \hbox{Definition of intersection} \cr & \iff & x \in A \land x \in B \land x \in A \land x \in C & \hbox{Definition of intersection} \cr & \iff & x \in A \land x \in B \land x \in C & \hbox{Simplification} \cr & \iff & x \in A \land x \in (B \cap C) & \hbox{Definition of intersection} \cr & \iff & x \in A \cap (B \cap C) & \hbox{Definition of intersection} \quad\halmos \cr}$$


Let us train our minds to desire what the situation demands. - Seneca


Contact information

Bruce Ikenaga's Home Page

Copyright 2017 by Bruce Ikenaga