Section 6.1, due on February 10

The principle of mathematical induction.  

What did I find difficult?

A number m in A is callde a least element (or a minimum or smallest element).

I've heard of induction in my past math classes - but I've never really understood the principle.  I cannot quantify how many points I've lost on proofs where the principle of induction could be used to show the proof to be true.  I was a little intimated with learning induction, due to my failure to understand it in the past.  That was what I found most difficult - overcoming my past fears of not knowing induction.  Now, this is what I understand induction to be:

For each positive integer n, let P(n) be a statement.  If 

1) P(1) is true and
2) the implication

If P(k), then P(k+1).

is true for every positive integer k, then P(n) is true for every positive integer n.

I understand a little more about what induction means.  I believe with more practice, I will understand it more.

What did I find interesting?

I did some more research on the principle of induction.  This is what I came up with - 

1. Show that it is true for the first one.

2. Show that if any one is true, then the next one is true.

It's interesting - every website I look on to learn more about the mathematical principle of induction teaches the principle using dominoes.  The domino effect is this:

1. the first domino falls.

2. if any domino falls, the next domino will fall

(thanks http://www.mathsisfun.com/algebra/mathematical-induction.html)

This makes more sense!  If the first domino falls, it will cause more dominoes to fall.  Eventually all dominoes will fall.  Understanding this makes the principle of induction hit home.  I mean, what kid hasn't built a line of dominoes, only to knock them down? If I can knock down the first one, then the next one will fall, and the next one and the next one and the next one.  Eventually, all of them will fall!