Set Theory masked with children and songs
Source: St Petersburg 2022 9.2
September 28, 2022
combinatorics
Problem Statement
schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of circle members can sing exactly songs, and any group of circle members - exactly songs. Prove that the group of all circle members can sing exactly songs.