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Set: {f(r,r):r in S}=S

Source: ISI (BS) 2007 #6

April 11, 2012
functionalgebra unsolvedalgebra

Problem Statement

Let S={1,2,⋯ ,n}S=\{1,2,\cdots ,n\} where nn is an odd integer. Let ff be a function defined on {(i,j):i∈S,j∈S}\{(i,j): i\in S, j \in S\} taking values in SS such that (i) f(s,r)=f(r,s)f(s,r)=f(r,s) for all r,s∈Sr,s \in S (ii) {f(r,s):s∈S}=S\{f(r,s): s\in S\}=S for all r∈Sr\in S
Show that {f(r,r):r∈S}=S\{f(r,r): r\in S\}=S
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