Solution to Find a system of distinct representatives for the following sets: {b,e,i,l}, {a,j,m}, {c,f,k}, {b,h,i,l}, {d,g,m}, … - Sikademy
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Archangel Macsika

Find a system of distinct representatives for the following sets: {b,e,i,l}, {a,j,m}, {c,f,k}, {b,h,i,l}, {d,g,m}, {e,h,k,l}, {a,d,j}, {g,j,m}, {c,e,k}, {a,g,j}, {f,h,i}, {d,j,k,m}, {b,c,f}.

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By Hall's Theorem,

An SDR (System of distinct representatives) is a collection of distinct elements a_{1}, a_{2}, \ldots a_{m}\ \text{such that}\ a_{i} \in A_{i}\ \text{for}\ i=1,2, \ldots m

Let A_{1}, A_{2}, \ldots, A_m be subsets of A

A_{1}, A_{2}, \ldots, A_{m} has an SDR if

\left|\bigcup_{i \in I} A_{i}\right| \geqslant |I|

In our case

Total no. of distinct elements =13 [b, e, i, l, a, j, m, c, f, k, h, d, g]

Total no. of sub sets =13

Hence,

System of distinct representatives,

=\left\{b, e, i, l, a, j, m, c, f, k, h, d, g\right\}


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Question ID: mtid-5-stid-8-sqid-1307-qpid-1045