Problem A. 791. (January 2021)
A. 791. A lightbulb is given that emits red, green or blue light and an infinite set \(\displaystyle S\) of switches, each with three positions labeled red, green and blue. We know the following:
\(\displaystyle i)\) For every combination of the switches the lighbulb emits a given color.
\(\displaystyle ii)\) If all switches are in a position with a given color, the lightbulb emits the same color.
\(\displaystyle iii)\) If there are two combinations of the switches where each switch is in a different position, the lightbulb emits a different color for the two combinations.
We create the following set \(\displaystyle U\) containing some of the subsets of \(\displaystyle S\): for each combination of the switches let us observe the color of the lightbulb, and put the set of those switches in \(\displaystyle U\) which are in the same position as the color of the lightbulb.
Prove that \(\displaystyle U\) is an ultrafilter on \(\displaystyle S\).
(\(\displaystyle U\) is an ultrafilter on \(\displaystyle S\) if it satisfies the following:
\(\displaystyle a)\) The empty set is not in \(\displaystyle U\).
\(\displaystyle b)\) If two sets are in \(\displaystyle U\), their intersection is also in \(\displaystyle U\).
\(\displaystyle c)\) If a set is in \(\displaystyle U\), every subset of \(\displaystyle S\) containing it are also in \(\displaystyle U\).
\(\displaystyle d)\) Considering a set and its complement in \(\displaystyle S\), exactly one of these sets is contained in \(\displaystyle U\).)
See also problem N.\(\displaystyle \,\)35. from the May issue of 1994 (in Hungarian).
(7 pont)
Deadline expired on February 15, 2021.
Statistics:
7 students sent a solution. 7 points: Fleiner Zsigmond, Füredi Erik Benjámin, Kovács 129 Tamás, Seres-Szabó Márton, Varga Boldizsár. 6 points: Sztranyák Gabriella. 1 point: 1 student.
Problems in Mathematics of KöMaL, January 2021