Problem:
If the operation x∗y is defined by x∗y=(x+1)(y+1)−1, then which one of the following is false?
Answer Choices:
A. x∗y=y∗x for all real x and y
B. x∗(y+z)=(x∗y)+(x∗z) for all real x,y, and z
C. (x−1)∗(x+1)=(x∗x)−1 for all real x
D. x∗0=x for all real x
E. x∗(y∗z)=(x∗y)∗z for all real x,y, and z
Solution:
x∗(y+z)(x∗y)+(x∗z)​=(x+1)(y+z+1)−1=[(x+1)(y+1)−1]+[(x+1)(z+1)−1]=(x+1)(y+z+2)−2​
Therefore, x∗(y+z)î€ =(x∗y)+(x∗z). The remaining choices can easily be shown to be true.