smc

Question

Let a @ b represent the operation on two numbers, a and b, which selects the larger of the two numbers, with = a. Let a ! b represent the operator which selects the smaller of the two numbers, with a ! a = a. Which of the following three rules is (are) correct? (1) = = a ! = (a ! b) @ (a ! c)$

Accepted Answer

The first rule must be correct as both sides of the equation pick the larger out of a and b. The second rule must also be correct as both sides would end up picking the largest out of a, b, and c. WLOG, lets assume b < c. The third rule is a little more complex. To see if it works, let's split the possibilities into three cases. a < b < c, b < a < c, and b < c < a. For the first case, the equation simplifies to a = a, which is correct. In the second case, the equation simplifies to a = a, which is correct. For the last case, the equation simplifies to c = c, which is also correct. Since all three cases work, this rule also works. Thus, our answer is all three