1. ## Set Counting Problem

The problem is:

In carrying out a survey of the efficiency of the lights, brakes and steering of motor vehicles, 100 vehicles were found defective as follows:

35 defective lights
8 had defective lights and brakes
7 had defective lights and steering
6 had defective brakes and steering

a) how many had defective lights, brakes and steering?
b) how many vehicles had defective lights only?

Now I know the answers are 5 and 25 respectively but I cannot form the correct equation to arrive at them.

I think the equation has to be in the form of

x(A u B u C)= x(A) + x(B) + x(C) - x(A n B) - x(A n C) - x(B n C) + (A n B n C)

I would appeciate help with the formulae.

Many thanks.

2. Hello, barc0de!

In carrying out a survey of the efficiency of the lights, brakes and steering of motor vehicles,
100 vehicles were found defective as follows:

35 defective lights
8 had defective lights and brakes
7 had defective lights and steering
6 had defective brakes and steering

a) How many had defective lights, brakes and steering?

b) How many vehicles had defective lights only?

Now I know the answers are 5 and 25 respectively,
but I cannot form the correct equation to arrive at them.

I think the equation has to be in the form of:
. . $\displaystyle x(A\cup B \cup C) \:= \:x(A) + x(B) + x(C) - x(A\cap B) - x(A \cap C)$ $\displaystyle -\; x(B \cap C) + (A \cap B \cap C)$
. . Right!

We are given:. . $\displaystyle \begin{array}{ccc} x(L\cup B\cup S) &=& 100 \\ x(L) &=& 35 \\ x(B) &=&40 \\x(S) &=&41 \\ x(L\cap B) &=&8 \\ x(L\cap S) &=& 7 \\ x(B \cap S) &=& 6 \end{array}\quad\hdots\quad\text{ and we want }x(L \cap B \cap S)$

Our equation is:

. . $\displaystyle x(L \cup B \cup S) \;=\;x(L) + x(B) + x(S) - x(L \cap B)$ $\displaystyle -\; x(L \cup S) - x(B \cap S) + x(L \cap B \cap S)$

. . . . . $\displaystyle 100 \qquad\;=\quad 35 \;+\; 40 \;\;+\;\; 41 \quad\;-\quad\;\; 8 \quad\;\; -\quad\;\;\; 7 \quad\;\;-\quad\;\;\; 6 \quad\;\;+$ $\displaystyle x(L \cap B \cap S)$

(a) And we have: .$\displaystyle 100\:=\:95 + x(L \cap B \cap S) \quad\Rightarrow\quad \boxed{x(L\cap B\cap S) \:=\:5}$

The best way solve part (b) is with a Venn diagram.

Or we can invent a formula for this situation . . .

We want: .$\displaystyle x(L \cap B' \cap S')$

And: .$\displaystyle x(L \cap B' \cap S') \;=\;x(L) - x(L \cap B) - x(L \cap S) + x(L \cap B \cap S)$

. . . . . . . . . . . . . .$\displaystyle = \;\;\; 35 \quad\;-\;\quad 8 \quad\;-\qquad 7 \quad\; +\qquad\;\; 5\quad=\quad\boxed{25}$