Question

Let R={0,60,120,180,240,360} and * is binary operation such that for a & b in R a*b is overall angular rotation corresponding to successive rotations by a and by b. Show that (R, *) is a group

Answer 1

Step-by-step explanation:

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Answer 2

We have shown that (R, *) is a group under the given operation.

To show that (R, *) is a group, we need to verify that it satisfies the four group axioms:

Closure: For any a, b in R, the operation a * b is also in R.

Associativity: For any a, b, and c in R, (a * b) * c = a * (b * c).

Identity: There exists an element e in R, such that for any a in R, a * e = e * a = a.

Inverse: 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.

Let's check each of these axioms:

Closure: If a and b are any elements of R, then a * b is also in R since it is also an angle measurement that is a multiple of 60 degrees.

Associativity: Let a, b, and c be any elements of R. Then we have:

(a * b) * c = (a+b) * c = (a+b+c) mod 360

a * (b * c) = a * (b+c) = (a+b+c) mod 360

Since the modular arithmetic operation is associative, we see that (a * b) * c = a * (b * c).

Identity: The identity element is 0, since 0+a = a+0 = a for all a in R.

Inverse: For any a in R, the inverse element is the additive inverse of a in Z/360Z, which is (360-a) mod 360. This is because a + (360-a) = (360-a) + a = 0 mod 360, which is the identity element in R.

Therefore, we have shown that (R, *) is a group under the given operation.

For such more questions on rotations,

https://brainly.in/question/29537846

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