HMMT 二月 2004 · 团队赛 · 第 12 题
HMMT February 2004 — Team Round — Problem 12
题目详情
- [25] Prove that there exists an element a ∈ S such that the equation a ? b = a holds for all b ∈ S . Sigma City [125 points]
解析
- Prove that there exists an element a ∈ S such that the equation a ? b = a holds for all b ∈ S . Solution: The same a as in the previous problem will do the trick. Indeed, for any b ∈ 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 this expression equals a ? ( b ? b ) = a , as required. Sigma City