Question

(a) Let R = {0, 60, 120, 180, 240, 300} and * = binary operation

so that for a and b in R, a * b is overall angular

rotation corresponding to successive rotation by a and by b.

Show (R, *) is a group. ​

Answer 1

Answer:

where are the options

Step-by-step explanation:

where are the options

Answer 2

Answer:

To show that (R, *) is a group, we need to verify four conditions:

1. Closure: For any a, b in R, a * b is also in R.
2. Associativity: For any a, b, c in R, (a * b) * c = a * (b * c).
3. Identity element: There exists an element e in R such that for any a in R, a * e = e * a = a.
4. Inverse element: For any a in R, there exists an element b in R such that a * b = b * a = e, where e is the identity element.

Firstly, to check closure, we need to show that a * b is also in R for any a, b in R. It is clear that any combination of two angles from R will give a new angle in R, so closure is satisfied.

Next, we need to show that the binary operation * is associative. For any a, b, c in R, (a * b) * c = (a + b + c) mod 360 = a + (b + c) mod 360 = a * (b * c), so associativity is satisfied.

The identity element e for * is 0, since a * 0 = 0 * a = a for any a in R.

Finally, we need to show that every element in R has an inverse element in R. Since the operation * corresponds to rotation, the inverse of any rotation by angle a is a rotation by angle (360 - a), which is also in R. Therefore, for any a in R, there exists an element b in R such that a * b = b * a = 0, where 0 is the identity element.

Since all four conditions are satisfied, we can conclude that (R, *) is a group.

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