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A4

Part of 1978 Putnam

Problems(1)

Putnam 1978 A4

Source: Putnam 1978

5/2/2022
A bypass operation on a set SS is a mapping B:S×S→SB: S\times S \rightarrow S with the property B(B(w,x),B(y,z))=B(w,z)B(B(w, x), B(y,z)) = B(w,z) for all w,x,y,z∈Sw, x, y, z \in S. (a) Prove that B(a,b)=cB(a,b)=c implies B(c,c)=cB(c,c)=c when BB is a bypass. (b) Prove that B(a,b)=cB(a,b)=c implies B(a,x)=B(c,x)B(a,x)=B(c,x) for all x∈Sx\in S when BB is a bypass. (c) Construct a bypass operation BB on a finite set S with the following three properties
[*] B(x,x)=xB(x,x)=x for all x∈Sx\in S. [*] There exist dd and ee in SS with B(d,e)=d≠e.B(d,e)=d \ne e. [*] There exist ff and gg in SS with B(f,g)≠f.B(f,g)\ne f.
PutnamBinary operation