Which of the following systems (G,o) are groups?

Question

Which of the following systems (G,) are groups?

(a) G is the set of all subsets of a set S, o is intersection.
(b) G is the set of all rotations about the origin of the real Euclidean plane, o is
composition of mappings.
(c) G = 2Z (the set of even integers), o is addition.
(d) G = 2Z, o is multiplication.
(e) G = Z, o is subtraction.

Answer 1

Check:
closure
associativity
identity element
inverse element

(a)
An intersection of subsets of G is itself a subset of G (closed under intersection)
(A*B)*C = (A*C)*(B*C) = A*C*(C*B) = A*C*C*B =
A*C*B = A*(C*B) associativity
G is the identity
G*A = A A*G = A
Inverse element?
Is there an element for A (call it D) such that A*D is G
no. Because A*D <= A
So, if A is not G, then intersecting anything with A will never give anything bigger than A, therefore you'll never get G.

b) Rotations I know are.

Let's us identify the rotations by the angle (in radians). Then it is a cyclical group (2*pi).
Any combination of rotations about the origin can be replaced with another rotation (closure)
The order in which the rotation are done does not affect the final result.
Any rotation of k*(2*pi), i.e., a full turn, is an identity. Rotate a figure by 8*pi and the figure has not changed (same as multiplying by 1 in arithmetic).
There exist a rotation of D=(2*pi)-A such that
D*A = identity

c)
closure? adding any number of even integers always yields an even integer.
associativity? the order of addition does not change the result
identity? 0 is an even integer and adding 0 does not change the element to which it is added.
inverse? For any even integer A, there exists an even integer -A.

d) no (find the problem)
e) by now, you know how to check

Answer 2

Write out all the characteristics of a group. Now go over them one by one for each question.

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