Say there are $46$ people in a club, $35$ of them are Chess lovers, $30$ of them are sports lovers, $40$ of them are Opera lovers, $38$ of them are video game lovers. So at least how many people are lovers for all $4$ activities?
I see there is a formula for questions like this. If the set size is $m$, with set $A, B, C, D...$ of size $a, b, c, d...$, Then:
minimum number for $A\cap B = a + b -m$
minimum number for $A\cap B \cap C = a + b + c- 2m$
minimum number for $A\cap B \cap C \cap D = a + b + c + d- 3m$
and so on.
So why is this? How can I understand this formula?