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Understanding Mathematical Induction

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The Domino Marketing System© consists of actions you take in a meaningful logical
order to accomplish a specific goal.
If you think of the system as lining up dominos all in a row, all you have to do is take
action to move the first domino forward, then everything else falls in place.
This is scientifically known as
"Mathematical Induction"
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The Principle of Mathematical Induction can be informally stated
by saying:
"We can establish the truth of a proposition if we can show that it
follows from smaller instances of the same proposition, as long as we can establish the
truth of the smallest instance (or instances) explicitly." (Grossman, 1990, p.
334)
In induction, the truth percolates up through the layers to prove
the whole proposition. If we know the first instance of a proposition is true and
the second one is derived from the first one, making it true, and the pattern repeats
itself, then we know that our final result will also be true. But if the first, or any of
the other instances following the first, turn out to be false then the final result will
be wrong.
Falling DominoesLet's look at a row of dominoes,
lined up and ready to be pushed over.

We know from experience that if we push over one domino, the rest of the dominoes
will fall over.

How can we prove this?
A. We know from experience that if we push over one domino, it
should fall over.

B. We also know that if a domino is falling
and has been placed correctly, it will knock over its neighbor.

Intuitively, it should be very clear that the fall should cascade all the way
up to the last domino. That is, if the next to the last domino falls, the last domino
also falls.

But now we need to think about increasing the rigor of this argument by doing a real
proof.
A Formal Proof
Let's look at the behavior of the dominoes.
- Assume that there's some domino k which doesn't fall over.
i.e. k is the first domino which doesn't work right (maybe it's out of line, too
far away, or has a different gravitational constant).
- Since k is the first such domino, the domino right before
k must have fallen over.
- But we know from B that a falling domino always knocks its neighbor
over.
- So domino k will fall over and we have a
contradiction.
What we have shown above is that because
We can knock over the first domino.
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and a falling domino knocks over its neighbor.

- then all the dominoes will fall over.
Now, if we think of each domino as an instance of a proposition, and if a given
instance is true, the corresponding domino will fall over.
If we can prove
The proposition is true in the first instance,
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And if a given instance is true,

The next one in the sequence will also be true.
- Then the proposition will be true in all instances.
This is called a Proof By Induction
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