Question

How many elements are in the union of four sets if
each of the sets has 100 elements,
each pair of the sets shares 50 elements,
each three of the sets share 25 elements,
and there are 5 elements in all four sets?

Answer 1

Answer:

20

Step-by-step explanation:

Let A, B, C and D sets.

Let |A| the number of elements of A.

Remember, the quantity of elements that have the set $A\cup B\cup C\cup D$ is

$|A\cup B\cup C\cup D|=|A|+ |B|+ |C|+|D|-|A\cap B|-|A\cap C|-|A\cap D|-|B\cap C|-|B\cap D|-|C\cap D|-|A\cap B\cap C|-|A\cap B\cap D|-|B\cap C\cap D|-|A\cap B\cap C \cap D|$

1. Since each set has 100 elements,

2.  each pair of the sets shares 50 elements

3. each three of the sets share 25 elements,

4. the four sets share 5 elements

Then

$|A\cup B\cup C\cup D|=100+100+100+100-50-50-50-50-50-50-25-25-25-5=20$

So, there are 20 elements in the union of the sets A, B, C and D