Problem:
Define xβ‘y to be β£xβyβ£ for all real numbers x and y. Which of the following statements is not true?
Answer Choices:
A. xβ‘y=yβ‘x for all x and y
B. 2(xβ‘y)=(2x)β‘(2y) for all x and y
C. xβ‘0=x for all x
D. xβ‘x=0 for all x
E. xβ‘y>0 if xξ =y
Solution:
For example, β1β‘0=β£β1β0β£=1ξ =β1. All the other statements are true:
(A) xβ‘y=β£xβyβ£=β£β(yβx)β£=β£yβxβ£=yβ΄x for all x and y.
(B) 2(xβ‘y)=2β£xβyβ£=β£2xβ2yβ£=(2x)β‘(2y) for all x and y.
(D) xβ‘x=β£xβxβ£=0 for all x.
(E) xβ‘y=β£xβyβ£>0 if xξ =y.
The problems on this page are the property of the MAA's American Mathematics Competitions