I am trying to solve a brainteaser and would like some direction if my line of thought is correct...
At a worker plant, 25 workers were absent on Monday, 22 absent on Tuesday and 19 absent on Wednesday. If none of the workers were absent on all three days, what is the smallest possible total number of workers absent on at least one day?
My initial thought is:
To find the smallest possible total number of workers absent on at least one day, we need to consider the overlap of absences between the days.
Let's denote:
$A$: Set of workers absent on Monday. $B$: Set of workers absent on Tuesday. $C$: Set of workers absent on Wednesday.
We are given: \begin{align*} |A| &= 25 \\ |B| &= 22 \\ |C| &= 19 \end{align*}
We need to find the smallest possible total number of workers absent on at least one day.
So $|A \cup B \cup C| = |A| + |B| + |C| - |(A \cap B)| - |(B \cap C)| - |(A \cap C)| + |(A \cap B \cap C)|$
Would I then be correct in assuming the maximum overlap happens on day 1 and 2, i.e. 22? Following this logic I end up with a value greater than any of the total absentees on one day which I believe to be wrong?