HMMT 二月 2015 · 冲刺赛 · 第 3 题
HMMT February 2015 — Guts Round — Problem 3
题目详情
英文原题
- [ 4 ] Find the number of pairs of union/intersection operations ( , ) ∈ {∪ , ∩} satisfying the fol-
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lowing condition: for any sets S, T , function f : S → T , and subsets X, Y, Z of S , we have equality ofsetsf ( X ) ( f ( Y ) f ( Z )) = f ( X ( Y Z )) ,
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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
英文解析
- [ 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 ofsetsf ( 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
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