CMSC 250 Homework 9 Fall 2001

Due Wed Oct 31 at the beginning of your discussion section.

You must write the solutions to the problems single-sided on your own lined paper, with all sheets stapled together, and with all answers written in sequential order or you will lose points.

1. Let $A = \{1,2,3,4,5\}$, $B = \{2,4,6,8,10\}$, $C = \{1,3,5,7,9\}$. $D=\{1,2\}$, $E=\{2,3\}$, $F=\{1,4\}$. The universal set $U= \{1,2,3,4,5,6,7,8,9,10\}$. Find the following:
   1. $A\cap B =$

   2. $B\cup C =$

   3. $A - C =$

   4. $C - A =$

   5. $D\times E =$

   6. $D\times E\times F =$

   7. $D\times (E\times F) =$

   8. $B^{c} =$

   9. $(B^{c})^{c} =$

2. Let $\Sigma = \{q,x,z\}$.
   1. $\Sigma^{0} =$

   2. $\Sigma^{1} =$

   3. $\Sigma^{2} =$

   4. Write at least 15 distinct elements of $\Sigma^{*}$.

3. $\Sigma = \{x,z,w,y\}$. Let $L$ be the set of all three-letter strings over $\Sigma$ that start with $x$ or $z$, end with $x$ or $y$, and have any letter in the middle so long as all three letters aren't the same.

4. Use Venn diagrams to show that the following statements are either true or false. Use two Venn diagrams for each part and show that the two diagrams are equal or different.
   1. $(A\cup B \cup C)^{c} = A^{c}\cap B^{c}\cap C^{c}$.

   2. $A\cap (B\cup C) = (A\cap B)\cup C$.

   3. $(A - B)\cup (B - A) = (A\cup B)-(A\cap B)$.

5. Let $A = \{A,E,F,H,I,K,L,M,N,T,V,W,X,Y\}$ Partition $A$ into subsets based on the number of straight lines it takes to write each of the letters. You may cross lines while drawing the letters, you may not change direction in any way when drawing a straight line.

6. Partition the integers from 2 to 20 into a set $P$ consisting of primes and a set $C$ consisting of composites.

7. Write the power set for $\{1,a,+, >\}$.

8. Prove the following statements using the set identities from chapter 5.2. Assume that $A$ and $B$ are subsets of some universal set $U$.
   1. If $A\subseteq B$, then $B^{c}\subseteq A^{c}$.

   2. $(A-B)^{c}= A^{c}\cup B$.

9. For all finite sets $E$, let $n(E)$ represent the number of elements in the set $E$, and let P$(E)$ denote the power set of $E$.

   Let $A$, $B$ and $C$ be three finite sets. Let $D = A\times B\times C$. Let $a\in A$, and $c\in C$, $b_{1}\in B$, $b_{2}\in B$, $b_{1}\neq b_{2}$, and that $\vert \sqrt{n({\bf {P}}(D))} - 12\vert < 2$, find $n(A)$, $n(B)$, and $n(C)$.

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