n(ξ) = 30, n(A) = 16, n(B) = 12, n(A ∩ B) = 5, find n((A ∪ B)′)

Step by step

The question

Question: n(ξ) = 30, n(A) = 16, n(B) = 12, n(A ∩ B) = 5.

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

5 are in both: they go in the middle first.

2 Fill the circles

In A: 16 in all, 5 of them are in the middle.

16 minus 5 is 11.

Careful!

The 5 in the middle are inside both circles.

Taking away 16 and 12 takes them away twice.

In B: 12 minus 5 is 7.

3 Find the outside

The circles hold 11 plus 5 plus 7: 23.

The rest of the 30 are outside: 7.

4 Check

Let's check: the four parts add up to 30.

So it fits!

Let's look back.

First: draw the Venn diagram.

Then: fill the circles.

Then: find the outside.

Last: check.