The operation between two symmetries can be thought of as a composition,with the result of the first symmetry being the starting point for thesecond symmetry. The composition symbol, °, will denote the operation between two symmetries. For example, the operation R180 ° MH would mean that, starting from the rightmost operation, the symmetry MH would be applied to the original figure first, with the result then being followed by R180, a counterclockwise rotation by 180 degrees.The result of the composition is equivalent to symmetry MV applied to the original figure (Figure 5), so we can express the final result as R180 ° MH = MV. Complete the following Cayley table:°Apply First Apply Second°1R180MHMV1 R180 MV MH MV The symmetries will be denoted by S, i.e., S = {1, R180, MH, MV}.ii. Explain why the system (S, °) is closed.iii. Provide at least three examples that can be used to conjecture that the system (S, °) is associative.iv. Identify the identity element for the system (S, °).v. For each element in S, specify their inverse element. Completing tasks (ii), (iii), (iv) and (v) indicate that the system (S, °) is a group.vi. Determine if the group (S, °) is commutative. The R180 symmetry is a rotation of the original figure, counterclockwise,by 180 degrees, as illustrated in Figure 3.ABDOFigure 3, The Riso SymmetryThe MH symmetry is a mirror reflection of the original figure about ahorizontal line through its center, as illustrated in Figure 4. Thissymmetry could also be denoted by Mo, the O indicating that thehorizontal line is at an angle of 0 degrees.ABABFigure 4, The MH or Mo SymmetryThe Mv symmetry is a mirror reflection of the original figure about avertical line through its center, as illustrated in Figure 5. This symmetrycould also be denoted by Mso, the 90 indicating that the vertical line is atan angle of 90 degrees.00ADCDFigure 5, The Mv or Mso SymmetryBAс Symmetry OperationsA symmetry of a figure can be thought of as a rigid motion that leaves thefigure unchanged. The figures that we'll consider are said to be finitefigures, and the rigid motions that we'll consider are rotations (in acounterclockwise or positive direction) and mirror reflections. Also, we'llsay that any finite figure has at least one symmetry, the "motion" of "doing-nothing" (or a rotation of zero degrees). We'll let 1 represent the "do-nothing" operation (which is also called the trivial symmetry), we'll let RArepresent a counterclockwise (positive) rotation of angle A, and we'll let Mrepresent a mirror reflection (using a different subscript to denote distinctmirror reflections).a. The figure shown below (Figure 1) has four symmetries that can bedenoted as 1, R180, MH and Mv.IFigure 1The symmetry, "1," is the "do-nothing" trivial symmetry, which can beillustrated as in Figure 2.ABFigure 2, The "1" SymmetryсBO

As you asked multiple subpart questions according to the rule I solved the only first three subpart,…

Answered: equivalent to symmetry MV applied to…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.2: The Law Of Cosines

Problem 35E

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Question

The operation between two symmetries can be thought of as a composition,with the result of the first symmetry being the starting point for thesecond symmetry. The composition symbol, °, will denote the operation between two symmetries. For example, the operation R180 ° MH would mean that, starting from the rightmost operation, the symmetry MH would be applied to the original figure first, with the result then being followed by R180, a counterclockwise rotation by 180 degrees.

The result of the composition is equivalent to symmetry MV applied to the original figure (Figure 5), so we can express the final result as R180 ° MH = MV.

Complete the following Cayley table:

°	Apply First
Apply Second	°	1	R₁₈₀	M_H	M_V
1
R₁₈₀			M_V
M_H
M_V

The symmetries will be denoted by S, i.e., S = {1, R180, MH, MV}.
ii. Explain why the system (S, °) is closed.
iii. Provide at least three examples that can be used to conjecture that the system (S, °) is associative.

iv. Identify the identity element for the system (S, °).

v. For each element in S, specify their inverse element. Completing tasks (ii), (iii), (iv) and (v) indicate that the system (S, °) is a group.

vi. Determine if the group (S, °) is commutative.

The R180 symmetry is a rotation of the original figure, counterclockwise,
by 180 degrees, as illustrated in Figure 3.
A
B
D
O
Figure 3, The Riso Symmetry
The MH symmetry is a mirror reflection of the original figure about a
horizontal line through its center, as illustrated in Figure 4. This
symmetry could also be denoted by Mo, the O indicating that the
horizontal line is at an angle of 0 degrees.
A
B
A
B
Figure 4, The MH or Mo Symmetry
The Mv symmetry is a mirror reflection of the original figure about a
vertical line through its center, as illustrated in Figure 5. This symmetry
could also be denoted by Mso, the 90 indicating that the vertical line is at
an angle of 90 degrees.
00
A
D
C
D
Figure 5, The Mv or Mso Symmetry
B
A
с

Symmetry Operations
A symmetry of a figure can be thought of as a rigid motion that leaves the
figure unchanged. The figures that we'll consider are said to be finite
figures, and the rigid motions that we'll consider are rotations (in a
counterclockwise or positive direction) and mirror reflections. Also, we'll
say that any finite figure has at least one symmetry, the "motion" of "doing-
nothing" (or a rotation of zero degrees). We'll let 1 represent the "do-
nothing" operation (which is also called the trivial symmetry), we'll let RA
represent a counterclockwise (positive) rotation of angle A, and we'll let M
represent a mirror reflection (using a different subscript to denote distinct
mirror reflections).
a. The figure shown below (Figure 1) has four symmetries that can be
denoted as 1, R180, MH and Mv.
I
Figure 1
The symmetry, "1," is the "do-nothing" trivial symmetry, which can be
illustrated as in Figure 2.
A
B
Figure 2, The "1" Symmetry
с
B
O

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