HMMT 二月 2004 · 团队赛 · 第 12 题
HMMT February 2004 — Team Round — Problem 12
题目详情
英文原题
- [25] Prove that there exists an element a ∈ S such that the equation a ? b = a holdsfor all b ∈ S .
Sigma City [125 points]
解析
英文解析
- Prove that there exists an element a ∈ S such that the equation a ? b = a holds for allb ∈ S .
Solution: The same a as in the previous problem will do the trick. Indeed, for anyb ∈ S , we have a ? b = ( b ? a ) ? ( b ? a ) = ( a ? a ) ? ( b ? b )
(we have used commutativity and associativity). But a ? a = a , and b ? b ∈ T , so thisexpression equals a ? ( b ? b ) = a , as required.
Sigma City