## POW 14

Pond Ivan.

My attempt…

After learning how to become a ninja (at a place up in Leonardstown) I tired to pond Ivan at any chance I could get. But soon became the weekend and I had no hope of finding him until…I walked in Starbucks. There he was sitting down drinking his coffee (and/or expresso) unexpecting of what was about to happen. I drew up multiple plans in my head: should I lead him away from his coffee by using complex math problems or should I grap him make and make him do my final then throw him in the pond. Too many thought were running through my head so I just ordered some starbucks left with a feeling of failure.

I will complete this POW.

The end.

## I am dying….

This weekend the Women’s SMCM Soccer team played at Catholic University, the reason why I am telling everyone this is because I want to share what our coach’s inspirational speech was to us: “Before you pass out you puke, before you die you pass out therefore you are not dying.” So while we are all dying with the amount of work we have, just think before we actually die we will puke and pass out from sleep deprivation first.

I am also getting excited about this scavenger hunt, it is becoming the talk of the campus (which isn’t really hard to do). From what Casey showed us the hint looks like the periodic table of Zelda. A few other of my classmates and I thought it would be a good idea if we all shared the location of the clues and where it leads on their blog so that it can be useful for the rest of the class. Also don’t forget to place the hint back after you find it so it doesnt mess everyone up.

Good luck on all your work this week! Get excited we only have three days of school left 🙂

## What are we doing with our lives?

So as I was looking through my notes to figure out what I should write about in my blog I kept coming across side notes saying look up Mantis Shrimp, therefore I did and hyperlinked the website to show us that we are going insane. But I must say this shrimp is pretty awesome and The Oatmeal does a great depiction on the mantis shrimp.

On another note, I think it would be interesting if we talked about why |(0,1) X (0,1)|=|(0,1)|. The first thing I do any problem like this is draw the two figures: a line segment from 0 to 1 and a cube with the dimensions from the origin (0,1). If we first look at the line segment we know that the segment 0 to 1 is part of the **R **which means that there are an infinite amount of numbers between (0,1). But if we take a segment from the line segment, 0 to .0001 we also know that interval is also **R** which also means there is an infinite amount of numbers. One way we determine that these figures all contain the same points is to prove that each of the figures are bijective (onto and one-to-one). The first question is we know that it is onto due to the shadow function, but how do we know it is one to one? The shadow function it alternating between digits after the decimal between x and y functions. For instance x=.12345678 and y=.2314678 the shadow function would look something like: 0.1231……etc.

Convex sets. scavenger

I really just think that this definition is pretty neat and I feel better knowing that this was a question on the previous exam, since everyone in our class pretty much understood the concept. Just to summarize since this a homework question, a convex set is when two points (x,y) and (a,b) lie in the sam e plane and can be connected by a line segment. One way to prove that a convex exist in a plane is to prove that the two points are equal to each other. Another way to prove a convex set is to let E=**R**^2. A nonconvex set is on a plane that has a cut out, in which two points can have a line segment that is not contained on the plain.

DON’T FORGET ABOUT OUR SCAVENGER HUNT THIS WEEK!!

s through FOM we only have 3 classes left…bitterssweet.

## New Normal

The other day during class Casey briefly mentioned this idea about the normal number, .123456789101112… I just though that it was so cool that this infinite number contains every possible set of numbers. For example the day you were born there is a sequence of numbers in that decimal that will have your exact date. Discovered or published by Champernowne is a transcendental constant otherwise writen as Csub10, the formula below expresses the normal number:

## To infinity and beyond

This week in class we talked about how there are multiple different types of infinity….mind blowing! Like most, I thought that there was only one type of infinity, that is a number that went on continuously, but now my world has been completely flipped up-side down. For instances the number line from [0,1] is uncountable– meaning that there is an infinite amount of numbers between that range. Another interesting fact that if the Reals are considered to be uncountable like the number line [0,1], but the natural number, also an infinite set, are considered to be countable. For example the number line [0,1] there are two number in the natural number line.When given two sets A and B one says they have the same cardinality by |A| =|B|, which also means that they are bijective. Any who infinity is pretty interesting and I can’t wait to learn about more infinities.

## M.I.A.

Sorry, I have not been posting for a while I have been going from doctor’s appointments to doctor’s appointments in all my free time (or blogging time). Therefore to makeup for my lack of posting I’m going to make this post really…really long and hopefully I will be able to cover everything we learned.

a) This weeks POW: kicks and hugs

In order for us to successfully complete this weeks POW our class decided to….

1) have the first person in line to count the number of “kicks” on everyone’s back

2) if there is an even amount of “kicks” the first person says “hugs” , but if there is an odd number of “kicks” the first person says kicks

3) after that it is up to the people in line to keep track of how many said “kick” or “hug”

AND we all get 100% 🙂

b) Equivalent Relations

For there to be an equivalent relation between to sets it must pass all three of the conditions: reflectivity, symmetry, and transitively.

Lets talk about Harry Potter….

example: Let s={students at Hogwarts}. Define s(1)~s(2) <=> s(1), s(2) elements of the same house

1) reflexive: Must be able to wiggle its self. For every s element of S, s~s. For this example: Harry Potter wiggles Harry Potter (check!)

2) symmetry: If it wiggles one-way it must wiggle the other way as well. For every a,b element of s , a~b => b~a. For this example: Harry Potter wiggles Ron implies Ron wiggles Harry Potter, since Harry and Ron are both in Gryff. this statement holds true. (check!)

3) transitively: There must be a third element in the set that wiggles both previous elements. For all a,b,c element of S such that a~b and b~c =>a~c. For this example: Harry wiggles Ron and Ron wiggles Hermione, implies that Harry wiggles Hermione (what a love triangle). This shows that the third element, Hermione, also belongs to the same house, Gryff. (check!)

Thus this is an example of an equivalent relation. But what is in it equivalent class??

c) Equivalent Classes

Denoted as s/~={equivalent classes)

From the example as before the equivalent classes is…

s/~={Gryff., Hufflepuff, Ravenclaw, Slytherin}

To represent an element in one of the classes use the symbol “[…]”. For example: [Neville] = [Harry Potter] element of s/~

d) Induction

STRONG induction <=> (WEAK) induction

Mind blowing weak induction and strong induction are actually the same thing and can be used interchangeably. Similar to contradiction and a orginal proof, these to methods can be used for the same proofs. It seems odd because it is harder(or seems impossible) to prove some theories using one method, but by using the other method it becomes simple.

I didn’t really elaborate very much on induction because I already have last weeks post about it, so check that one out.

Thanks for reading, plus I do believe Casey told the 1:20 class to call out Brad for missing class and for ultimately missing a STRONG induction proof example.

## Lets play dominoes!

Yesterday in class we learned about Poof by Inductions. This proof technquie allows us to prove infinitely many statements. The key to this proof is to relate it to dominoes. Proof by induction is always preformed in two steps. The first step is the base case, in relationship to dominoes it is knocking down the first domino in the series. The second step in proof by induction is the inductive step. Think of this step as all the dominoes being lines up if one falls they all fall. To conclude when in doubt remember dominoes.

## A few things…

Last class we were introduced to something that we wished we all knew before the exam…

THE BIG QUESTIONS:

1) To prove something, what am I requires to do?

2) What am I allowed to use?

Since Casey shared the “BIG QUESTIONS” I thought it would be helpful to share some of my techniques to writing proofs. The best thing technique that I found useful before writing a proof is to list out what relates to the set we were given. I go as far as writing the simple definitions that we learned in the begining of the semester to help me understand the set better. Also if I am ever stuck on the problem I realize the best thing is to step away and try it again with fresh eyes, because most of the time you see something that you had not seen before.

I also wanted to discuss some of chapter 4.2…

statement: p =>q

Contrapositive: ~q => ~p

Converse: q => p

Inverse: ~p => ~q

The orginial staement and the contrapostive are equivalent.

When p => q and q =>p , we write that as p <=> q. Where the symbol <=> is represents the phrase if and only if.

I hope this was helpful!