Download Domino Theory. Domino theory refers to a

Domino Theory.
Domino theory refers to a demonstration of a standing line of dominos
knocking each other down.
First, we line the dominos up:
Then we tip the first domino:
And all of the dominos fall down:
The corresponding idea in mathematical logic is called Mathematical
Induction, where we use a sequence of mathematical theorems instead of
dominos.
For Mathematical Induction we perform two steps:
1. First we line the dominos up. We must prove that one theorem being
true implies that the next theorem is true. This lines the theorems up
like dominos. This is called the induction step.
2. We tip the first domino: We prove the first theorem. This is called the
trivial step.
At this point we are finished, and consider all of the theorems proved.
(Just like we believe all of the dominos fall down).
For example, here is a proof of the binomial theorem for positive integer
powers:
The sequence of theorems we wish to prove is this:
For every positive integer n,
1. Lining the dominos up:
Suppose for some number n the formula is true. We then wish to
show that the formula is true for n+1.
We have this equation:
So the coefficient of
is given by
Expanding these coefficients gives n!/(n-i)!i! + n!/(n+1-i)!(i-1)!
n! factors out ,1/(i-1)! Factors out, and 1/(n-i)! factors out giving
[n!/((n-i)!(i-1)!)][(1/i)+(1/(n+1-i))].
Adding (1/i)+(1/(n+1-i)) gives (n+1-I+i)/(i(n+1-i)) = (n+1)/(i(n+1-i)).
Moving the n+1 into n! gives (n+1)!.
Moving the 1/i into 1/(i-1)! gives 1/i!.
Moving the 1/(n+1-i) into 1/(n-i)! gives 1/(n+1-i)!
Combining these results we get (n+1)!/((n+1-i)!(i!)) which equals
.
Thus we have
This completes the induction step.
We plug 1 in for n to get the trivial step:
Since this is true, we have performed both parts of the Mathematical
induction for the sequence of theorems and we can say that
is true for all positive integers n.