I have to prove this by induction:
$n$ cars are travelling down a narrow one-way street. We know that:
- The distance d between each two cars is the same.
- The safe breaking distance b is the minimum distance between two cars that is needed for the second car to stop on time if the car in front suddenly breaks.
- d < b
Prove by induction or refute: if the first car suddenly stops moving, all cars will stop moving. Before you do the induction state the property P you are using in the induction axiom.
So this is what I have until now and please tell me if it is right or wrong.
The property P: if $n=1$ stops, then $n=2$ will stop. (Is this property correct?)
We consider 2 cases:
Base case:
n=1 -> Since there is one only car on the road, that car will stop. (Should the base case start at 0, 1 or 2? )
Step case: (Supposed that the base case should start at 1)
As mentioned, for n=1 if the only car stops, then we can say all the cars have stopped.
Assume that all cars stop for n cars and then we prove that all cars stop for $n+1$. We create the set C of Cars in which are included all the cars in road. The moment the first car $(n=1)$ stops, the following cars have 2 ways to stop:
- $n+1$ car stops without touching the n car
- $n+1$ car stops by touching and having/or not an accident with n car
Any suggestion?
When the first car stops, the second cannot stop on time, as $d < b$. Then the statement "all cars will stop moving" is false.
This holds whatever $n$, except for $n=1$.