# Minimum and maximum number of elements in intersection of sets.

In a battle 70% of the combatants lost one eye, 80% an ear, 85% a leg, 75% an arm, x% lost all four limbs.Then

• What is the minimum value of x?

• What is the maximum value of x?

• From van diagram I figured that the minimum value of x is 10%. But I am not satisfied with my version of explanation.



• I think maximum will be 70% – Kangkan Sep 12 '17 at 6:52
• wait does all 4 limbs mean losing 1 ear, 1 arm and 1 leg? – Joao Noch Sep 12 '17 at 8:47
• One eye, one ear, one leg and one arm – Kangkan Sep 12 '17 at 9:43

## 2 Answers

You're right about max, because it's $\min(70\%,75\%,80\%,85\%)=70%$.

For the next question (which you were right as well) I find it easier to explain with numbers, say out of $100$.

The minimum intersection of $70$ and $75$ is $70-(100-75)=70-25=45$

The minimum intersection of $80$ and $85$ is $80-(100-85)=80-15=65$

The minimum intersection of $45$ and $65$ is... (hint: same way as the previous two) $10$.

• So what's the generalization to this? sum(SET) - (len(SET) - 1) * 100? At least, that's what I inferred from this video – zelusp Oct 22 '19 at 20:46

I believe the general algorithm to determine the minimum is as follows (in Python)

def get_min_intersection (slices,total = 100):
if sum(slices)- (len(slices)-1) * total >= 0:
return sum(slices)- (len(slices)-1) * total
else:
return 0

# The minimum intersection of [70,80,75,85] out of 100 is 10
assert get_min_intersection([70,80,75,85]) == 10

# The minimum intersection of [90,60,80] out of 100 is 10 according to https://youtu.be/GcT4ExbSYRI?t=34
assert get_min_intersection([90,60,80]) == 30

# The minimum intersection of [30,35,32] out of 50 is 0 according to https://youtu.be/GcT4ExbSYRI?t=251
assert get_min_intersection( [30,35,32], 50) == 0

# The minimum intersection of [40,50,60] out of 100 is 0 according to https://www.handakafunda.com/set-theory-maximum-and-minimum-values/
assert(get_min_intersection( [40,50,60]) ==  0)
`

As @Joao said, the maximum intersection is the minimum of the elements in the set.