X * y = x +y + 1 (i) x * y = x + y + 1: y * x = y + x + 1 = x + y+ 1. (Thus, * is commutative.) (ii) x * (y * z) = x * (y + z + 1) = x + (y + z + 1) + l = x + y + z + 2. (x * y) * z = (x + y + l) * z = (x + y + l) + z + 1 = x + y + z + 2. (* is associative.) (iii) Solve x * e = x for e: x * e = x + e + 1 = x: therefore, e = - 1. Check x (- l) = x + (- l) + l = x;(- 1) * x = (-l) + x + 1 = x. Therefore, - 1 is the identity element. (* has an identity element.) (iv) Solve x * x' = - 1 for x': x * x' = x + x' + 1 = - 1: therefore x' = - x - 2. Cheek: x * (-x - 2) = x + (-x - 2) + 1 = - 1: (-x - 2) * x = (- x - 2) + x + 1 = - 1. Therefore, -x - 2 is the inverse of x. (Every element has an inverse.) x * y = x + 2y + 4 (i) x * y = x + 2y + 4: y * x = (ii) x*(y * z) = x*() = (x * y) * z = ()* z = (iii) Solve x * e x for e. Check. (iv) Solve x * x' = e for x'. Check. x * y = x + 2y - xy x * y = |x + y| x * y = |x - y| x * y = xy + 1 x * y = max {x, y} = the larger of the two numbers x and y