Problem:
If the operation xβy is defined by xβy=(x+1)(y+1)β1, then which one of the following is false?
Answer Choices:
A. xβy=yβx for all real x and y
B. xβ(y+z)=(xβy)+(xβz) for all real x,y, and z
C. (xβ1)β(x+1)=(xβx)β1 for all real x
D. xβ0=x for all real x
E. xβ(yβz)=(xβy)βz for all real x,y, and z
Solution:
xβ(y+z)(xβy)+(xβz)β=(x+1)(y+z+1)β1=[(x+1)(y+1)β1]+[(x+1)(z+1)β1]=(x+1)(y+z+2)β2β
Therefore, xβ(y+z)ξ =(xβy)+(xβz). The remaining choices can easily be shown to be true.