CS 281 Solutions to Homework 7

pp. 54 - 55

2 a) $A \cap B$, b) $A \cap \overline{B}$, c) $A \cup B$, d) $\overline{A} \cup \overline{B}$

6 (b) Suppose that $A \cap \emptyset$ $\neq$ $\emptyset$.
Then there is an element $c$ that is in $A \cap \emptyset$, i.e. $c \in A \cap \emptyset$.
Hence $c \in A$ and $c \in \emptyset$. But $c \in \emptyset$ can not be true.
Hence $A \cap \emptyset$ $\neq$ $\emptyset$ is not true.
Hence $A \cap \emptyset$ = $\emptyset$.

(f) Since for any $x$ $x \in U$ is true, $x \in A \cup U$ $\rightarrow x \in U$ is true.
Hence $A \cup U \subseteq U$.
Also since for any $x$ if $x \in U$ is true, $x \in A \vee x \in U$ is true.
Hence if $x \in U$, then :w $x \in A \cup U$.
Hence $U \subseteq A \cup U$ is true.
Hence $A \cup U = U$.

8. $A = \{1,5,7,8,3,6,9\}$, $B = \{2,10,3,6,9\}$

10 (d) $A \cap (B-A) = A \cap ( B \cap \overline{A} )$
= $( A \cap \overline{A} ) \cap B$
= $\emptyset \cap B$ (property 12 of set operations)
= $\emptyset$

16. First $(A - B) - C$ = $( A \cap \overline{B} ) \cap \overline{C}$. Then
$(A - C) - (B - C) = (A \cap \overline{C}) \cap (\overline{B \cap \overline{C}} )$
= $(A \cap \overline{C}) \cap ( \overline{B} \cup C )$
= $A \cap ( (\overline{B} \cap \overline{C} ) \cup (\overline{C} \cap C) )$
= $A \cap ( (\overline{B} \cap \overline{C} ) \cup \emptyset )$
= $A \cap (\overline{B} \cap \overline{C} )$ = $(A - B) - C$.

Textbook p. 210
22 a) Let $O$ be the set of odd integers.
Basis Clause: $1 \in O$
Inductive Clause: If $x \in O$, then $(x + 2) \in O$.
Extremal Clause: Nothing is in $O$ unless it is obtained from the Basis and Inductive Clauses.

b) Let $T$ be the set of positive integer powers of $3$.
Basis Clause: $3 \in T$.
Inductive Clause: If $x \in T$, then $3x \in T$.
Extremal Clause: Nothing is in $T$ unless it is obtained from the Basis and Inductive Clauses.

c) Let $P$ be the set of polynomials with integer coefficients.
Basis Clause: $1 \in P$, and $x \in P$.
Inductive Clause: If $p \in P$ and $q \in P$, then $(p + q) \in P$, $(p-q) \in P$ and $pq \in P$.
Extremal Clause: Nothing is in $P$ unless it is obtained from the Basis and Inductive Clauses.