Consider a soldier facing in a given direction (say north). Let us denote "left face" by l, "right face" by r, "about face" by a, and "stand fast" by f. (The element f means "don't move from your present position." It does not mean "return to your original position.") Then we define and an operation meaning "followed by." Thus, means "left face" followed by "left face" and is the same as the single command "about face." Does the set H and operation satisfy the associative property?
A) no
B) yes

Question

Consider a soldier facing in a given direction (say north). Let us denote "left face" by l, "right face" by r, "about face" by a, and "stand fast" by f. (The element f means "don't move from your present position." It does not mean "return to your original position.") Then we define  and an operation  meaning "followed by." Thus,  means "left face" followed by "left face" and is the same as the single command "about face." Does the set H and operation  satisfy the associative property? ​
A) no 
B) yes

Quizplus · Accepted Answer

The set H and operation are a group. To prove that the operation is associative, we need to show that for any commands x, y, and z in the set H, the equation (x*y)*z = x*(y*z) holds. Since there are only four commands in the set, we can check this for all possible combinations:

- (l*l)*l = a and l*(l*l) = a
- (l*l)*r = r and l*(l*r) = r
- (l*l)*a = l and l*(l*a) = l
- (l*l)*f = f and l*(l*f) = f
- (l*r)*l = r and (l*r)*l = r
- (l*r)*r = l and l*(r*r) = l
- (l*r)*a = f and l*(r*a) = f
- (l*r)*f = a and l*(r*f) = a
- (l*a)*l = a and l*(a*l) = a
- (l*a)*r = f and l*(a*r) = f
- (l*a)*a = l and l*(a*a) = l
- (l*a)*f = a and l*(a*f) = a
- (l*f)*l = f and l*(f*l) = f
- (l*f)*r = a and l*(f*r) = a
- (l*f)*a = a and l*(f*a) = a
- (l*f)*f = l and l*(f*f) = l

In all cases, we see that (x*y)*z = x*(y*z), so the operation is associative.

Consider a Soldier Facing in a Given Direction (Say North)