HMMT 二月 2015 · 冲刺赛 · 第 3 题

HMMT February 2015 — Guts Round — Problem 3

[ 4 ] Find the number of pairs of union/intersection operations ( , ) ∈ {∪ , ∩} satisfying the fol- 1 2 lowing condition: for any sets S, T , function f : S → T , and subsets X, Y, Z of S , we have equality of sets f ( X ) ( f ( Y ) f ( Z )) = f ( X ( Y Z )) , 1 2 1 2 where f ( X ) denotes the image of X : the set { f ( x ) : x ∈ X } , which is a subset of T . The images f ( Y ) (of Y ) and f ( Z ) (of Z ) are similarly defined.

解析

[ 4 ] Find the number of pairs of union/intersection operations ( , ) ∈ {∪ , ∩} satisfying the fol- 1 2 lowing condition: for any sets S, T , function f : S → T , and subsets X, Y, Z of S , we have equality of sets f ( X ) ( f ( Y ) f ( Z )) = f ( X ( Y Z )) , 1 2 1 2 where f ( X ) denotes the image of X : the set { f ( x ) : x ∈ X } , which is a subset of T . The images f ( Y ) (of Y ) and f ( Z ) (of Z ) are similarly defined. Answer: 1 If and only if = = ∪ . See http://math.stackexchange.com/questions/359693/overview-of-b 1 2