When Casey gives my blog a shout-out on his blog:

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If…Then Statements: Problems from Class!

1. Take the abstract mathematical statement represented here with variables: P ⇒ Q. 
Under what conditions is this statement taken to be true? False?

 To find the answer, go here! The only other information you need is that P represents the hypothesis and Q represents the conclusion.

2. Five more mathematical statements to define! (Define P, Q, and give relationship based on what the speaker means)

A. If it’s sunny in the morning, I cycle to work. 

P: If it’s sunny in the morning

Q: Then I cycle to work

Relationship: The sunny morning is causing this person to cycle to work

B: If you eat your peas, then you get ice cream!

P: If you eat your peas

Q: Then you get ice cream

Relationship: The fact that you eat your peas (what a noble venture, by the way), is causing you to get the (awesome) reward of ice cream

C: When Gotham is ashes, you have my permission to die.

First of all, how very morbid.

P: If Gotham at some point becomes ashes

Q: Then you will have Bane’s permission to die (Yay….!)

Relationship: Gotham going up in flames is giving cause to your death.

D: If you elect me, your taxes will go down!

P: If you elect me

Q: Then taxes will decrease (ahahahaha)

Relationship: Electing someone will be the causality for taxes decreasing

E: If a number has only itself and one as factors, we the call the number prime.

P: If a number has only itself and one as factors

Q: Then that number is considered prime

Relationship: The conditions of a prime number being prime gives cause to calling that number prime.

3. Find the truth set for “p is an even prime”.

{2}

4. Find the truth set for “The function f(x) = x2 + 2x + a has two roots”

To find roots for an equation, we can use the quadratic formula! The formula, for the majority of us who don’t remember algebra two, is:3ea647783b5121989cd87ca3bb558916

Luckily, we already know what a and b are, so we can plug them into the equation, simplify, and get the equation:

-1±√1+c

Roots can be real or unreal, so having c be a negative number is not an issue. The only issue we run into is when c = -1. At that point, the equation only has one root. So, the truth set is:

{all real numbers where c ≠ -1.}

5. Find the truth set for “A ⊆ {1,2,3,4,5} and |A| ≤ 2”

First of all, the notation |A| simply means that you are counting up the number of elements in a set. So the truth set must contain sets that have two elements each or less.

{{  },{1},{2},{3},{4},{5},{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}}

This set includes the empty set, sets with only one number from the A subset, and sets with 2 numbers from the A set.

6. Find the truth set for  “A,B⊆ {1,2,3,4,5} with A⊆B and |B| – |A| = 4”

To unpack this meaning, first we realize that A and B are both sets made up of the numbers 1, 2, 3, 4 and/or 5. The second part of this open sentence is saying that A is a subset of B, which means that all elements of A are in B as well. Finally, the last part means that the set B must either have 5 or 4 elements in it, with 1 or 0 elements in A, respectively. So this truth set will be pretty large…

{ (A = { }, B={1,2,3,4}) – and all other possible combinations of 4 elements of A & B, where A is the empty set.

(A= {1}, B={1,2,3,4,5}) – and all other possible combinations of A equaling different elements from the subset.}

And now for the most important part:

BONUS: “Strong men also cry.” What movie is this from??

The Big Lebowski. Obviously.

 

Class Review: Conditional Statements, Open Sentences Continued (Kind of)

So I’ve already painstakingly articulated A LOT about conditional statements, which we went over in class today. So instead of outlining my notes again, I thought I’d focus on one thing I was confused about in class that I want to make sure I get right: The difference between being an element and being a subset. The example that confused me a little was:

S ⊆ ℤ (where S is a variable set)

The truth set would have to be comprised of sets itself, NOT just integers. An integer is an element of the set that is a subset of all integers. The proper way to explain the truth set for this open sentence would be {all sets of integers}, NOT {all integers}. Integers are elements, sets of integers are a subset to all integers.

Make sense? Think so.

FOM BOOK: 1.3

The Introduction Section starts out by explaining proofs, which is basically the root of all of FOM (hehe, math humor). To write a good proof, you must do the following:

  1. Assert a hypothesis and derive a conclusion (If, then statement) 
  2. Interpret the hypothesis based on the problem at hand
  3. Work out the rest of the proof with the interpretation

Get Your Hands Dirty #1

If X is an even integer, and Y is a multiple of X, then Y is even.

First of all, since x is even, it can be rewritten as x = 2n, so Y is a multiple of 2n.

When the problem says that Y is a multiple of X, that means that Y is multiplied by X over and over to create a set of numbers. Say that x=2. Y would then be a multiple of 2, meaning that multiples produced would include 2Y, 4Y, 6Y, 8Y, etc. Multiples produced with just X and Y are XY, 2XY, 3XY, 4XY, etc.

So, since Y is a multiple of 2n (which equals x), Y’s multiples are 2NY, 4NY, 6NY, etc. Because there is a two included in each multiple, each of Y’s multiples in this case are even.

Since Y must be multiplied by 2N in each of it’s multiples, Y will always be even.

Get Your Hands Dirty #2 (Go here to read about conditional statements first!)

A. The sum of two real numbers is always real.

If x and y are both real, then x+y is always real. 

The hypothesis states that x and y must be real, and the conclusion from that is that the sum of x and y is always real.

B. A nonempty set has at least two subsets.

If a set S is nonempty, then S has at least two subsets. 

The hypothesis states that we must first be talking about a nonempty set, and the conclusion from this is that specific nonempty set must then have at least two subsets.

C. Every nonvertical line has a y-intercept.

If a line, y=mx+b, is nonvertical, then y=mx+b must have a y-intercept. 

The hypothesis states the condition that a line must be nonvertical to fit the conclusion that it will always have a y-intercept.

Open Sentencesstatements that are not known to be true or false, as the sentence contains nonspecific variables.

Substitution/Replacement: replacing a variable in an open statement with a specific object.

Truth Value: the characterization of a statement as either true or false

Truth Set: set of values that make an open sentence true

Universe of Discourse: boundaries from which replacement values are chosen (ex: universe = all reals; U = generic universe)

Get Your Hands Dirty #3:

A. 0 < x < 10 (universe = Z

{1,2,3,4,5,6,7,8,9}

B. (t-2)(t^2 – 9) = 0 (universe = R)

{-3, 2, 3)

C. B is a subset of {2,4,6,8,10} with exactly two distinct elements (universe = the set of subsets of {2,4,6,8,10})

For B to be a subset of {2,4,6,8,10}, then B must be either the number 2,4,6,8, or 10. It also must have two distinct elements, which means that each element in the truth set must include two, non repeating numbers mentioned in the set already. The truth set will be really (relatively) large then, including:

{ {2,4}, {2,6}, {2,8}, {2,10}, {4,6}, {4,8}, {4,10}, {6,8},{8,10}, {6,10} }

Relating Subsets and Conditional Sentences: If a conditional statement is true, the truth set for the hypothesis should be a subset of the conclusion. Anything that makes the hypothesis true should then make the conclusion true. Written another way:

\subseteq \!\, B  means “if x \in \!\,A, then x\in \!\,B”

Counterexample:  a value that makes a hypothesis true and a conclusion false. (To read more about what makes conditional statement true or false, go here!)

Even number: an integer that has 2 as a factor; two-directional meaning – all even integers have a factor of two and all numbers with a factor of two are even

IT ALL MAKES SENSE.

Class Review: Conditional Statements – the most confusing thing you will ever think about (not really)

So a conditional statement is basically a formula to set up a hypothesis and a conclusion. The basic formula is

IF (hypothesis), THEN (conclusion).

These statements do not need to be true to be considered a conditional statement. For example, the statement If Casey likes teaching FOM, then we will all get A’s, is a conditional statement, even though it is (unfortunately) not true.

How do we know then if these statements are true or false? Simple.

To make a conditional statement true:

  • Any hypothesis that is true should be followed by any conclusion that is true. So, a statement like If I like pancakes, then my name is Kate. They don’t necessarily have to be connected in any way, both parts just must be true. It seems weird, but logical causality is different from straight-forward causality. 
  • Any hypothesis that is false that is followed by any sort of conclusion. So, a statement like If the Earth is flat, I’ll legally change my name to Beyoncé. It doesn’t matter what the conclusion is, because the condition of this statement will never be proven true, and therefore the conclusion will never be used. This condition is vacuously true. 

To prove a conditional statement false:

  • The hypothesis must be true, while the conclusion proves false. For example, a statement like If I am in a FOM class, I am a math major. The conclusion is false because I am a math minor, but I am in a FOM class. Therefore the hypothesis is true, but not the conclusion. 

With this knowledge, I want to talk about the problem we were given in class today that I’m still trying to figure out. The problem is:

You see four cards and are told that each one of them has a letter on one side and a number on the other. You see the four cards as pictured here:

A 3 T 8

Which of the four cards must you turn over to verify the claim:

“On the opposite side of every vowel is an even number”

First, we’ll make it a conditional statement. If a card has a vowel on one side, then there will be an even number on the other side. 

So to prove this right or wrong, we must focus on the conclusion, not the hypothesis. If we were to focus on the hypothesis and prove that wrong, we would be focusing on a whole other statement, not the one at hand. We don’t need to worry about what happens when the card doesn’t have a vowel on one side, we’re focusing on what happens when it does, so we’ll make the condition that the hypothesis is true.

When the hypothesis is true, the statement is true if the conclusion is true, and false if the statement is false. Casey also mentioned in class that it is much easier to prove something false than true. To prove something false, you only need to bring up one instance that proves it false, but to prove something unquestionably true, you would need to flip all the cards.

Or would we? Let’s say we focus on only vowels. T would then be irrelevant to the statement all together because it is not mentioned in the hypothesis. To prove this statement right then, we would have to flip both the 8 card and the A card, to see that there is a vowel on the other side of the 8 card and an even number on the other side of the A card.

To prove this statement wrong, we would need to flip A to see if there is an odd number, and flip 3 to see if there is a vowel on the other side. If either of these cases happened, the statement would be false.

It seems to me that A is the crucial card. Either way, no matter if you are trying to prove it true or prove it false, you need to flip A to see what is on the other side.

Whew, that was confusing. I hope it makes any sort of sense.

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Class Review: Sets

On Friday we talked about sets in class.

Definition

A set is just a collection of things, mathematical or otherwise. I could have a set of numbers, symbols, ice cream cones, pandas, whatever.

Say I have a set of: {7, 42, #, h, @, $} Each of the characters in this set is called an element. Since this set also has numbers, symbols, and letters in it, we can also call it a set of mixed types.

Some Common Sets

N = all natural numbers

Z = all integers

Q = all rational numbers

R = all real numbers

These are not the only sets in existence, just the most common.

Another common set we talked about was the null or empty set, which is exactly what you think it is. The set with nothing in it. It’s written as : {   }.

The set {∅} does not mean the same thing. That set is not empty, because it has another set inside of it: the null set.

Set Relationships

Sets can be related to each other as well. For example, all natural numbers are integers. Therefore, the natural numbers are a subset of integers. Similarly, all integers are rational, so integers are a subset of rationals. All rational numbers are real, so rational numbers are a subset of real numbers.

Then we bring complex numbers into the mix. Complex numbers are all real numbers..plus the unreal numbers. This means that we can say that real numbers are a subset of complex numbers, as all real numbers fall into the category of complex. We can write out what complex numbers are made of as:

{a + (i)b: a,b \in \!\, \mathbb{R} \!\,}, so when we’re talking about a real number, b=0.

Union/Intersection

A union of two sets means:

\cup \!\, B = {x: x \in \!\, A or x\in \!\,B}

Which basically means that this new set contains elements of A or B, it doesn’t matter where they come from.

An intersection means:

\cap \!\,B = {x: x \in \!\, A and x\in \!\,B}

Which basically means that this new set only contains elements that are in both A and B.

When you subtract two sets, the new set becomes:

A – B = {x: x \in \!\, A and x∉B}, which basically just a mathy version of realizing that when you subtract two sets, the new set is all of the elements in A, with all elements of B taken away.

You can compare more complicated sets as well. Say we compare 2×2 matrices and all real numbers.

M2x2 \cap \!\, \mathbb{R} \!\, = ∅, because the two sets have nothing in common, so their intersection set contains nothing. The 2×2 matrices may contain real numbers, but the sets are in two different formats, and can therefore not be compared.

Theorems

If B \subseteq \!\, A, then B \cap \!\, A = B.

If B \subseteq \!\, A, then B \cup \!\, A = A.

When you think about it, these two theorems make sense.

If B is a subset of A, that means that all elements in B are elements in A as well, so when only looking at elements in common with both A and B, that is simply all of B.

Since the same holds true for B being a subset of A, and we compare the two sets so that we use all elements part of A and B, this set would just be A, because A is comprised of all of B’s elements plus some unique elements only in A.

Infinity

The cool thing about intersection and union is the fact that you can perform both an infinite amount of times.

For example, if you wanted to compare these sets:

[-1 , 1] \cap \!\, [-1/2 , 1/2] \cap \!\, [-1/4 , 1/4]….(to infinity and beyond)

You would look at what all of these sets that are getting infinitely smaller, and see what elements they share. In these case, this intersection would equal the set {0}, since the sets are getting smaller and smaller.

Quotes on quotes on quotes

William Thurston

“The product of mathematics is clarity and understanding. … The real satisfaction from mathematics is in learning from others and sharing with others. “

To put it lightly, this guy was pretty freaking smart. At one point in the 70’s, professors advised students not to go into Thurston’s line of mathematics after college, because he was solving too many theorems for there to be any work left for them to do by the time they graduated. First off, he worked with geometric things called foliations, which basically have to do with these things called manifolds, which have to do with these other things called tangent bundles (see what I mean? his EARLY work can’t even be understand by someone without a PhD). Thurston’s shining moment was probably his geometrization conjecture, which led to many other conjectures in math. The conjecture revolved around 3-manifolds, which are topological figures where each point is 3 dimensional, and the fact that they can be broken down into pieces, which each follow one of the eight  geometric structures.

            Whatever that means. (Shout out to this article for breaking the conjecture down to at least a little bit of common english!)

Charles Hermite

“We are servants rather than masters in mathematics.”

He was also a brainiac, a French one at that. He solved the quintic equation, which, according to Wikipedia, looks like this:

g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,

I’m going to have to agree with Charles here, and admit that this equation pretty much has the upper hand to me in a math fight.

Blaise Pascal

“To speak freely of mathematics, I find it the highest exercise of the spirit; but at the same time I know that it is so useless that I make little distinction between a man who is only a mathematician and a common artisan. Also, I call it the most beautiful profession in the world; but it is only a profession.”

Blaise was actually a genius. He dabbled in philosophy, religion, science, and mathematics. By the age of 16, he had already published a paper on conics. He created and manufactured his own calculation machine, is said by his sister to have “taught himself geometry”, and worked with probability theory.

Hermann Weyl

“God exists since mathematics is consistent, and the devil exists since its consistency cannot be proved.”

Besides being a pretty clever guy, Hermann casually connected mathematics with theoretical physics. He basically took someone else’s equations on electromagnetic fields, and connected it with geometry by way of describing it’s geometric properties in space-time. This opened up a lot more dialogue in the field of differential geometry. Hermann also worked a lot with group theory, specifically talking about atoms through the view of groups in matrices. This breakthrough formed quantum theory.

You go mathematicians, paving the way for math-lovers everywhere!

The Goldbach Conjecture

The conjecture states:

Every even integer greater than 2 is the sum of 2 primes.

How do we prove it? Nobody really has that figured out yet.

It’s simple to try every single number individually to see if this conjecture holds true, but that does not give proof that it is true for every single number ever in existence. Many mathematicians have spent time pushing the bounds of this conjecture to higher and higher numbers, and have even tried to prove a weaker version of this conjecture.

The weaker version states that all odd numbers greater than nine are the sum of 3 odd primes. The strong version states that all even integers greater than 4 can be broken down into the sum of two primes. The Goldbach Partition is then written as: 2n = p + q, where n is a positive integer and p and q are both prime numbers.

Terence Tao proved that every integer greater than 1 is the sum of no more than 5 prime numbers, which is a step in the right direction to proving this conjecture true. Vinogradov has proven that all sufficiently large odd numbers work out to be a sum of 3 primes, and Estermann has proven that almost all even numbers are the sum of two primes.

So far, no one has completely proven Goldbach’s true conjecture.

 

“Mathematics is a game played according to certain simple rules with meaningless marks on paper.”

I would have to agree with this statement. At first it seemed a little condescending to me, just because the quote is calling the concept of writing out math meaningless. Upon more thought, I realized that it makes sense. Math isn’t supposed to mean only what is on paper. Math represents ideas, which are easier to explain when written down.

I fully agree with the game aspect of this quote. It’s a fun game too! You can go about solving problems in many different ways, but most math requires a creative mind, and an ability to think of many ways to try a single problem. This is especially true with the problems we were working on in class today. I felt like I was playing a game! I thought I would finally have one problem figured out, someone else would say something, or prove something that puts me back at square one (much like Uno, which is my favorite card game). And it’s always fun to get the right answer before anyone else, let’s be honest.

Math is pretty simple as well, as long as you know the rules. The rules aren’t hard, they just build on each other, making it hard for some people who don’t remember the first rules. For example, if you don’t remember how to solve an equation for x, you’re going to have a hard time in any math class after algebra. The rules aren’t hard, just very important.

In my math classes so far, there has always been a common goal to solve a problem, and I’ve always found it entertaining. Before being set free to try and solve a problem, my teachers have always broken down how to do it, or given me exclusions, examples, or exceptions so I don’t get completely lost. So to think of math as a game with simple rules makes sense to me.

This quote is from David Hilbert, who was a 19th century mathematician known for simplifying geometry down to a series of simple statements, and was one of the “founding fathers” of formal foundations of mathematics (get it?! because this is for foundations of mathematics..the class..).

So yes, I think this quote is a very fun and accurate way to sum up math as a whole.