Problem
COM-B2-M08-P006 Three Clubs and Three Students
#6
★★★☆☆ Level 3 of 5
Three clubs have possible leaders \(A_1=\{a,b\}\), \(A_2=\{b,c\}\), \(A_3=\{a,c\}\). Prove that distinct leaders can be chosen for all clubs.
Check Hall's condition for subsets of clubs.
For one club there are \(2\) options. For any two clubs, the union of lists contains all three names or at least two names. For all three clubs, the union is \(\{a,b,c\}\), of size \(3\). Hall's condition holds, so an SDR exists. For example, \(A_1\to a\), \(A_2\to b\), \(A_3\to c\).
First explicit Hall application.