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addition table on $X$

Source: 3-rd Taiwanese Mathematical Olympiad 1994

January 16, 2007
functionvectoralgebra unsolvedalgebra

Problem Statement

Given X={0,a,b,c}X=\{0,a,b,c\}, let M(X)={f∣f:X→X}M(X)=\{f|f: X\to X\} denote the set of all functions from XX into itself. An addition table on XX is given us follows: ++ 00 aa bb cc 00 00 aa bb cc aa aa 00 cc bb bb bb cc 00 aa cc cc bb aa 00 a)If S={f∈M(X)∣f(x+y+x)=f(x)+f(y)+f(x)∀x,y∈X}S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}, find ∣S∣|S|. b)If I={f∈M(X)∣f(x+x)=f(x)+f(x)∀x∈X}I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}, find ∣I∣|I|.
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