Abstract algebraic operations

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Problems:

(1) Show that the operation * on the real number is defined by a*b = ab + a2 is neither associative nor commutative.

(2) Show that * on R defined by a*b = a+b+2 is both associative and commutative

(3) The movements of a robot are restricted to no change (N), turn left (L), turn right (R), and turn about (A). Construct a Cayley Table and show that this set of movements forms a group.

Solutions:

(1) Associativity: check whether or not (a*b)*c = a*(b*c)

(a*b)*c = (ab + a2)*c = (ab + a2)c + (ab + a2)2 = a(b + a)c + [a(b + a)]2 ( 1)
Using the properties of addition and multiplication of real numbers (i.e. commutativity, associativity and distributivity), one can write:
(a*b)*c = ac(a + b) + a2(a + b)2 = a(a + b)[c + a(a + b)] = a(a + b)(a2 + ab + c) ( 2)
a*(b*c) = a*(bc + b2) = ...