Step by step
The question
Question: n(ξ) = 24, n(A) = 10, n(B) = 9, n(A ∩ B) = 4.
Find n((A ∪ B)′).
n of A means how many are in A.
A ∩ B means in both.
(A ∪ B)′ means in neither.
1 Draw the Venn diagram
4 are in both: they go in the middle first.
2 Fill the circles
In A: 10 in all, 4 of them are in the middle.
10 minus 4 is 6.
in A only = 10 − 4 = 6
Careful!
The 4 in the middle are inside both circles.
Taking away 10 and 9 takes them away twice.
Careful: not 24 − 10 − 9 = 5 ✗
In B: 9 minus 4 is 5.
in B only = 9 − 4 = 5
3 Find the outside
The circles hold 6 plus 4 plus 5: 15.
The rest of the 24 are outside: 9.
neither = 24 − (6 + 4 + 5) = 9
4 Check
Let's check: the four parts add up to 24.
So it fits!
6 + 4 + 5 + 9 = 24
Let's look back.
First: draw the Venn diagram.
Then: fill the circles.
Then: find the outside.
Last: check.
Answer
neither = 24 − (6 + 4 + 5) = 9