# find the intersection value of 3 subsets

#### schinb65

##### New member
Let a, b and c be three subsets of universe U with the following properties: n(A)= 63, n(B)=91, n(c)=44, The intersection of (A&B)= 25, The intersection of (A&C)=23, The intersection of (C&B)=21, n(A U B U C)= 139. Find the intersection of (A&B&C).

I am told the answer is 10.

I tried drawing a diagram.

A: B: C:
x x x
25-x 21-x 23-x
23-x 25-x 21-x
63-(x + 21-x + 23-x) 91-(x+25-x+21-x) 44-(x+23-x+21-x)

I added all of these together and had them equal to 198, since I am using values repeatedly. I also omitted the repeats and set the sum to 139. This should give me the correct answer correct?

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Let a, b and c be three subsets of universe U with the following properties: n(A)= 63, n(B)=91, n(c)=44, The intersection of (A&B)= 25, The intersection of (A&C)=23, The intersection of (C&B)=21, n(A U B U C)= 139. Find the intersection of (A&B&C).

I am told the answer is 10.

I tried drawing a diagram.

A: B: C:
x x x
25-x 21-x 23-x
23-x 25-x 21-x
63-(x + 21-x + 23-x) 91-(x+25-x+21-x) 44-(x+23-x+21-x)

I added all of these together and had them equal to 198, since I am using values repeatedly. I also omitted the repeats and set the sum to 139. This should give me the correct answer correct?
What you have written in the last paragraph is essentially what you need to do. I don't understand the rest of your attempt though but I think you are on the right track. You might find this useful Inclusion

#### soroban

##### Well-known member
Hello, schinb65!

Let $A, B, C$ be three subsets of universe $U$ with the following properties:

. . $\begin{array}{c}n(A)\,=\, 63 \\ n(B)\,=\,91 \\ n(C)\,=\,44\end{array} \qquad \begin{array}{c} n(A\cap B) \,=\,25 \\ n(B\cap C) \,=\,21 \\ n(A\cap C) \,=\, 23 \end{array} \qquad n(A\cup B \cup C) \,=\,139$

Find: $n(A \cap B\cap C)$

Are you familiar with this formula?

$n(A \cup B \cup C) \:=\:n(A) + n(B) + n(C)$

. . . . . . . . . . . . . . $+\,n(A \cap B) + n(B \cap C) + n(A \cap C)$

. . . . . . . . . . . . . . . . $+ n(A \cap B \cap C)$

#### Jameson

Staff member
Hello, schinb65!

Are you familiar with this formula?

$n(A \cup B \cup C) \:=\:n(A) + n(B) + n(C)$

. . . . . . . . . . . . . . $+\,n(A \cap B) + n(B \cap C) + n(A \cap C)$

. . . . . . . . . . . . . . . . $+ n(A \cap B \cap C)$
Hi soroban,

$n(A \cup B \cup C) \:=\:n(A) + n(B) + n(C) -\,n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$.