 |
|
|
|
|
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:
|
|
|
| |
|
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? |
|
|
