Vtyjkyuk

In the book by K.D.Joshi, titled 'Foundations of Discrete Math.', there is given on page#66, a partial set theoretic approach to solving the above problem.

The book states the condition as : there are a set of $5$ locks with $L_i$ can be opened by person $p_i$, i.e first person ($p_1$) able to open lock $L_1$.

The book states the problem requirement as : Union of any three of these five subsets be the whole set $L$, while the union of any two need not be the whole set $L$.

The book further states the problem soln. in terms of the De-Morgan laws as :
For each $i$, let $M_i$ be the complement of $L_i$ in $L$.

Then the problem amounts to finding a suitable set $L$ & some subsets $M_1, M_2,cdots, M_5$ of $L$ s.t.

(i) for any $i$ and $j$, $M_i ∩ M_jne 0$,

(ii) for any three distinct $i,j,k,$ $ M_i ∩ M_j ∩ M_k = 0$.

I have $2$ confusions.

1 . Think that the author wanted a set-theoretic formulation, for no explicit purpose. I hope that only something practical can be got, as here.

2 . The problem does not seem to be fully solved, if the theoretic rep. is taken alone. But, can only express some possible hierarchy of locks, say $L_1 + L_2 +L_3 =1$, any other combination of two locks $=0. $

Want to explore what the set notn. is intended for in direct, or complement form.

On the linked page the author does not go into the actual solution of the problem (this is deferred to a later section); he only rewrites the given conditions in terms of complements. He indeed does not tell why. My conjecture is that the actual solution will be simpler to describe in terms of the complements.