Session 9, Homework:
Homework

Problem H1

In this game, starting with a string of Ys and Zs, the object is to simplify the string by following strict rules. The rules are:

 • YYY can be erased. • ZZ can be erased. • The commutative law holds: YZ = ZY. • E is the empty string (a string with no Ys or Zs).

 Example 1: Step 1: YZZYYZYZYYZ (first erase ZZ) Step 2: Y YYZYZYYZ (erase YYY) Step 3: ZYZYYZ (commute YZ) Step 4: ZZYYYZ (erase ZZ and YYY) Step 5: Z (can't be simplified)
 Example 2: Step 1: ZYYYZ (erase YYY) Step 2: ZZ (erase ZZ) Step 3: E (empty string is left)

Simplify the following strings:

 a. YZYZZYYZ b. YYYYZZYZY c. YZYZYZYZYZYZYZYZZZYZYZYYZY

 Problem H2 Including the empty string E, there are six essentially different strings that cannot be simplified. They are called the elements of the YZ group. Find all the elements of the YZ group.

Problem H3

The symbol "*" represents the operation "put together and simplify." For example:

 • YZ * YZ = YY • Y * E = Y

Compute:

 a. E * YZ b. YZ * YY c. Z * YZ

Problem H4

Find the missing term.

 a. YZ * __ = E b. Z * __ = YZ c. YY * __ = Z

 Problem H5 For the YZ group, * works a little bit like multiplication. Another way to write the first two rules is Y3 = E and Z2 = E. Explain.

 Problem H6 The only powers of Y are Y, Y2, and E. Explain.

 Problem H7 Find all the powers of each element of the YZ group.

Problem H8

Simplify:

 a. Y1,000 b. (YZ) 1,001

 Problem H9 Make a * table.

 Problem H10 What element of the group works like the number 1 for multiplication?

 Problem H11 What is the reciprocal of each element?

 Homework problems are adapted from Algebra: Themes, Tools and Concepts, by Anita Wah and Henry Picciotto (New York: Creative Publication, Wright Group/McGraw-Hill, 1993), p. 159. The above materials may not be reproduced without the written permission of Wright Group/McGraw-Hill. The above materials may not be reproduced without the written permission of Wright Group/McGraw-Hill.
 Session 9: Index | Notes | Solutions | Video