Monthly Archives: February 2013

This blew my mind today in class and I thought I might as well blog about it.  I thought it was really cool how negating something to the millionth time would actually just be  the first negation. Therefore this means that we could figure out how to negate things to the trillionth. This is possible because negation runs in cycles similar to imginary numbers.  Mind blown. 

On another note I was interested in Cyrus question today about  Fibonacci and golden numbers so i check out this website (the one highlighted in blue) to get a better explanaion. Anywho its is actually really cool and I recommend checking it out, in fact I recommend checking out the entire website it is about Fibonacci and Nature.

Advertisements

More Functions

To continue from last post….functions!!

A function f: A–>B

Surjective=onto=EPIC (can be used interchangably)

A surjective function means every element in B is an output for f (the function).

Example of surjective function: 

   y=x^3  

x^3

 

Injective=one-to-one=monic

A injective function means for every input there is a single output (aka: the horizontal line test)

Example of  injective function:

y=e^x

e^x

 

Bijecture

A bijecture function means a function that is both epic and one-to-one

Example of bijecture function:

y=3x+5

y=3x+5

 

 

 

Prisoners of FOM

FUNCTIONS:

Yesterday in class Casey talked about functions, I thought that it would be beneficial to reenforce some of the concepts.

The first concept that I thought is important is that for every input there is an output. This means that for a function to exist it must pass the vertical line test. Thus for every input there must be a single output.

images

Gentle Definition:
A function from a set A to a set B is a rule that associates some b/in B to elements a/in A (latex fail)

The second conceept that I thought to reenforce would be the hardcore definition that Casey explained in class. The reason why “math people” like the hardcore definition is mainly because it is easier to place in a proof. It has teo elements in the defintion that must satisfied before the function is consider a true function (aka pass the vertical line test)

To conclude, don’t forget to do this week’s homework and worksheet on functions!!!

POW #5

Helping my roomie with vector!!

Helping my roomie with vector!!

POW #4

coins

10 coins in a bag, each having a different value, consecutively:

1,2,3,4,5,6,7,8,9, and 10 dollar coins

We know that a minimum $10 of coins will be donated…

(1,9) (2,8) (3,7) (4,6) (10)

to at least 5 charities. But If we only donated minimum of $10 to each charitiy we would not be using all the coins, in fact the (5) coin is missing. Supicious, we try donating $11 to each charitiy by using ALL the coins.

TRY $11:

(1,10) (2,9) (3,8) (4,7) (5,6)

We still serve 5 charities, but would be giving each charity more money and use all the coins!

So does a PATTERN emerge?

9 coins—-> Each charity gets $9 each
10 coins—> Each charity gets $11 each
11 coins—> Each charity gets $11 each
12 coins—> Each charity gets $13 each

In terms of n:

Odd number of coins n is an element of 2Z+1, (n-1)/2 charities would recieve $n each

Even number of coins n is an element of 2Z, (n)/2 charities would recieve $n+1 each

“Die Hard” doing FOM

The other day in class Casey told us to look up the Die Hard: Jug Problem, so what else was I suppose to do except look it up. In short, the jug problem has to do with basic addition and subtraction. The two men each have a container, one container holds 5 gallons while the other one holds 3 gallons. In order for them to get 4 gallons to disarm the bomb they do the following mathematical operations:

5-3=2 (Poured the full 5 gallon jug into the three gallon jug, leaving only two gallons in the 5 gallon jug)

3-2=1 (Poured te 2 gallons from the 5 gallon jug and empty the 3 gallons into the 5 gallon jug)

5-1=4 (Fill up the 5 gallon jug, then fill up the 3 gallon jug (containing two gallons of water) from the five gallon jug, leaving 4 gallons in the five gallon jug)

At first I thought it was confusing, but once you go through it slowly it make complete sense! This trick it actually really cool and useful if I ever need to disarm a bomb with exactly 4 gallons of water. Thanks to Bruce Willis for saving the day!

Music to my ears

I just wanted to mention that last semester I attended Professor Kung’s presentation at St. Mary’s Hall about music and math’s relationship. I thought the presentation was by far the most interesting lecture I have been to thus far. The musicians playing at the presentations were very talented and had great stage presence. Professor Kung’s presentation was basically about the wavelengths of musical instruments and how he could capture the sound than analyze it significance using advance technology. He uses props to show how the scientific finding which allows for less advance math students to follow. To conclude I recommend seeing his presentation it was AMAZING!!!!

It’s a miracle

Following up on my last post, yestrday inclass we learned that the conjecture if |s|=n then |P(s)|=2^n. Ultimately this means that the answer to number seven on the in-class worksheet is 65,536.
Example: 2^2=4
2^4=16
2^16= 65,536

{{{{{{{{{{{{????????????}}}}}}}}}}}}

In Class worksheet:
Question #7 {{{{{{{{{???????????}}}}}}}}}}}

My group and I after taking the second power set of this question were overwhelmed with the amount of {0} and after spending almost the entire class period figuring out this problem all came up with different answers. So in conclusion, I ask all fellow FOM students if they have any idea to form an answer for question number 7 without taking all the power sets?

OUR WORK SO FAR FOR QUESTION #7:
((0)= null set)

P({0})= {0,{0}}—>2
P(P({0}))={0,{0},{{0}},{0,{0}}}—->4
P(P(P({0})))={0,{0},{{0}},{{{0}}},{{0,{0}}},{0,{0}},{0,{{0}}},{0,{0,{0}}},{{0},{{0}}},{{0},{0,{0}}},{{{0}},{0,{0}}},{0,{0},{{0}}},{0,{0},{0,{0}}},{{0},{{0}},{0,{0}}},{0,{0},{{0}},{0,{0}}}}—->12

IDEAS:
~could you possibly say since the first power set has two elements the rest are powers of 2^x
ie. 2^1=2
2^2=4
2^3=16—–> so are we missing four possible matches :/

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{HELP!!!!!!}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

Getting down and dirty

Frist off the Super Bowl is today, which means Ravens with beat the 49ers!!!!! Any who, I thought it would be most beneficial for myself as well to others in FOM to summarize this week’s material.

On Monday, we learned about conditional sentences, statements of the form “if___,then__”. The first section of the conditional sentence is the hypothesis which is followed by the conclusion. In conditional sentences the order matters the hypothesis must always come before the conclusion. The nerdy notation for conditional statements is: H=>C. The arrow looking figured, =>, means implies. So for example, (f(X) differentiable) => (f(x)is continuous).

To continue on Wednesday we learned about open sentences, the truth set, and the “universe”. To begin, an open sentence has a variable,x, and what we substitute for x changes then sentence’s truthiness. In short its a sentence that is sometimes true sometimes false. Th “universe”= {all possible things you can plug-in to your variables}. And lastly a truth set= {all substitutions that make the open sentence true}.

Finally Friday, we continued to learn about truth sets/counterexamples as well as power set of S and the cartesian product. Counterexample are objects that makes the conclusion to a conditional sentence false. The cartesian product is written as AxB. An example of the cartesian product which forms a new set is A={1,2,3} and B={4,5} equals AxB= {(1,4),(2,4),(3,4),(1,5),(2,5),(3,5)}. Lastly the power set of S is the set of all subsets of S. Example, S=0={} and P (0)={0}.

To conclude…GO RAVENS!!!