Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 A B C DHomework

Solutions for Session 9, Part H

See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6| H7| H8| H9| H10| H11

Problem H1

a:

 Step 1: YZYZZYYZ Step 2: YZYYYZ (erase ZZ) Step 3: YZZ (erase YYY) Step 4: Y (erase ZZ)

b:

 Step 1: YYYYZZYZY Step 2: YZZYZY (erase YYY) Step 3: YYZY (erase ZZ) Step 4: YYYZ (commute last ZY) Step 5: Z (erase YYY)

c:

 Step 1: YZYZYZYZYZYZYZYZZZYZYZYYZY (better think more systematically) Step 2: YYYYYYYYYYYYYZZZZZZZZZZZZZ (commute all Ys first, Zs last) Step 3: YZ (erase 12 Ys by threes, 12 Zs by twos)

 Problem H2 The elements of the YZ group are E, Y, YY, Z, YZ, and YYZ.

Problem H3

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

Note that in all of these, any three occurrences of Y can be removed, as can any two occurrences of Z. Since the commutative law exists for this group, order is not important.

Problem H4

 a. YZ * YYZ = E b. Z * Y = YZ c. YY * YZ = Z

 Problem H5 Y3 means the same as Y * Y * Y, which is YYY, which is the same as E. The same is true of Z2, which is identical to Z * Z.

 Problem H6 Since Y3 is identical to E, Y4 will be identical to Y, Y5 = YY, Y6 = E, etc.

Problem H7

Powers of each element:

 • E: E only • Y: Y1 = Y, Y2 = YY, Y3 = E • YY: YY1 = YY, YY2 = Y, Y3 = E • Z: Z1 = Z, Z2 = E • YZ: YZ1 = YZ, YZ2 = YY, YZ3 = Z, YZ4 = Y, YZ5 = YYZ, YZ6 = E • YYZ: YYZ1 = YYZ, YYZ2 = Y, YYZ3 = Z, YYZ4 = YY, YYZ5 = YZ, YYZ6 = E

Problem H8

 a. Y1,000 = Y1, since 1,000 = 1 (mod 3), and the powers of Y repeat every three powers. b. (YZ)1,001 = (YZ)5, since 1,001 = 5 (mod 6), and the powers of YZ repeat every six powers. According to the list of powers of YZ, (YZ)5 = YYZ. Another way to do this is to imagine a line of 1,001 Ys and 1,001 Zs, and decide what would be left after all the cancellation.

 Problem H9 Here is the completed table: Note that every element appears exactly once in each row, and once in each column.

 Problem H10 The element E works this way, since E * A = A for any element A in the table, just like 1 * N = N for any number N.

Problem H11

The reciprocals can be found by finding E within the row and column of each element. Here are the reciprocals, in pairs:

 • E and E • Y and YY • Z and Z • YZ and YYZ

 Session 9: Index | Notes | Solutions | Video