	Various Proof Techniques
Back Up	Some of you may be familiar with proof techniques. Things like Proof by Contradiction, Proof by Contrapositive, Proof by Cases, and Proof by Induction are tools that all Mathematicians employ to make proving things easier or possible. You may be less familiar with these.. Proof by Sadism - When a book says something like "We'll leave it to the reader to show.." or "It is a standard exercise to prove that.." This basically means figure it out on your own. Proof by Masochism - Similar to Proof by Sadism except it is self-imposed. I can't turn this page till I convince myself of this minor detail! Proof by Vigorous Hand Waving - Invoked by many professors by stating a Theorem and then proceeding to glance over a proof while simultaneously waving his hands violently. This gesture is to symbolize to the student that he knows what he is talking about. Proof by Boxing - It's in a box! It must be correct! Some books separate out important Theorems into their own little box. Proof by Colored Chalk - All Theorems will now be marked in Red.. Means that you should copy down anything in Red if you happen to be taking notes. Proof by Example - The proof is too difficult to be done in one lecture, so let me show you this cute little example instead. Proof by Reference - I could show you the details of the proof, but it's all in your book. Look it up on your own time! Proof by Exam - Oh I didn't get time to prove it in class, so I'll just make you guys prove it by putting it on the next exam! Proof by Problem Set - Same as above except there is no way you would be able to prove it in 3 hours, so instead I assigned it as problem 1 on your next problem set. Proof by Prose - Instead of showing why this is true mathematically, imagine you are the vector. Wouldn't you be all alone if you were by yourself on the plane? Therefore, another vector, which we will call the companion vector, must exist. Clearly.. Proof by Challenge - Truth is we don't know if this Conjecture is true, so problem 2 on your problem set is to Prove or Disprove it. Good luck.. Proof by Assumption - Assuming this the Theorem follows effortlessly. Unfortunately, that assumption is not always kosher. Proof by Deferral - We'll postpone the proof of this Theorem to the end of this Chapter to avoid breaking the flow of the arguement. Proof by Trying Everything - If we try this.. No wait that turns ugly fast. Hmm.. maybe if we tighten this restriction. Oh wait that works. Proof by Luck - Wow, it is a good thing I applied the operator before summing up all the terms because now the expression reduces greatly! Proof by Dream - It was amazing. I fell asleep thinking about this proof and finally realized the key missing element when I woke up! Proof by Praying - I've been working on the Proof for weeks! Please God help me! Proof by Copying - You know this friend of yours who is a Math God.. I'm sure he would be happy to explain to you why your answer is wrong. Proof by Epiphany - Oh I'm such an idiot! Now it is so simple! Why didn't I realize this before?! Proof by Strong Words - Clearly, proposition A is true. Therefore, as a consequence, proposition B must be true as well. As a result, it's obvious to the most casual observer that the original proposition is true. QED Proof by Verboseness - Let us call this point P. This point P can also be represented as a vector, which we will also denote P. Any ambiguity should be settled by examining the context in which P appears. If neccessay we may denote the vector as P' to avoid confusion. Continuing with the arguement, this point (or vector) P was chosen arbitrarily so.. Proof by Faith - Trust me, it's True. We'll just take it as a given for the rest of this arguement. Proof by Defeat - Eh the Proof is good enough. I just want to turn in this Problem Set and be done with it! Proof by Recall - You were suppose to have already learned this proof in the less advanced course, so I won't waste any time proving it. Proof by Reconstruction - Hmm.. I'm suppose to get from here to there. Looks like I need to assume this here and oh yeah he mentioned in class that we might need the Fundamental Theorem of Algebra so I'll throw that in there. Oh wait there is a gap here. Lessee.. Assuming this earlier in the arguement fixes that gap. Basta! QED Now onto number 3..

Various Proof Techniques

Some of you may be familiar with proof techniques. Things like Proof by Contradiction, Proof by Contrapositive, Proof by Cases, and Proof by Induction are tools that all Mathematicians employ to make proving things easier or possible. You may be less familiar with these..

Proof by Sadism - When a book says something like "We'll leave it to the reader to show.." or "It is a standard exercise to prove that.." This basically means figure it out on your own.
Proof by Masochism - Similar to Proof by Sadism except it is self-imposed. I can't turn this page till I convince myself of this minor detail!
Proof by Vigorous Hand Waving - Invoked by many professors by stating a Theorem and then proceeding to glance over a proof while simultaneously waving his hands violently. This gesture is to symbolize to the student that he knows what he is talking about.
Proof by Boxing - It's in a box! It must be correct! Some books separate out important Theorems into their own little box.
Proof by Colored Chalk - All Theorems will now be marked in Red.. Means that you should copy down anything in Red if you happen to be taking notes.
Proof by Example - The proof is too difficult to be done in one lecture, so let me show you this cute little example instead.
Proof by Reference - I could show you the details of the proof, but it's all in your book. Look it up on your own time!
Proof by Exam - Oh I didn't get time to prove it in class, so I'll just make you guys prove it by putting it on the next exam!
Proof by Problem Set - Same as above except there is no way you would be able to prove it in 3 hours, so instead I assigned it as problem 1 on your next problem set.
Proof by Prose - Instead of showing why this is true mathematically, imagine you are the vector. Wouldn't you be all alone if you were by yourself on the plane? Therefore, another vector, which we will call the companion vector, must exist. Clearly..
Proof by Challenge - Truth is we don't know if this Conjecture is true, so problem 2 on your problem set is to Prove or Disprove it. Good luck..
Proof by Assumption - Assuming this the Theorem follows effortlessly. Unfortunately, that assumption is not always kosher.
Proof by Deferral - We'll postpone the proof of this Theorem to the end of this Chapter to avoid breaking the flow of the arguement.
Proof by Trying Everything - If we try this.. No wait that turns ugly fast. Hmm.. maybe if we tighten this restriction. Oh wait that works.
Proof by Luck - Wow, it is a good thing I applied the operator before summing up all the terms because now the expression reduces greatly!
Proof by Dream - It was amazing. I fell asleep thinking about this proof and finally realized the key missing element when I woke up!
Proof by Praying - I've been working on the Proof for weeks! Please God help me!
Proof by Copying - You know this friend of yours who is a Math God.. I'm sure he would be happy to explain to you why your answer is wrong.
Proof by Epiphany - Oh I'm such an idiot! Now it is so simple! Why didn't I realize this before?!
Proof by Strong Words - Clearly, proposition A is true. Therefore, as a consequence, proposition B must be true as well. As a result, it's obvious to the most casual observer that the original proposition is true. QED
Proof by Verboseness - Let us call this point P. This point P can also be represented as a vector, which we will also denote P. Any ambiguity should be settled by examining the context in which P appears. If neccessay we may denote the vector as P' to avoid confusion. Continuing with the arguement, this point (or vector) P was chosen arbitrarily so..
Proof by Faith - Trust me, it's True. We'll just take it as a given for the rest of this arguement.
Proof by Defeat - Eh the Proof is good enough. I just want to turn in this Problem Set and be done with it!
Proof by Recall - You were suppose to have already learned this proof in the less advanced course, so I won't waste any time proving it.
Proof by Reconstruction - Hmm.. I'm suppose to get from here to there. Looks like I need to assume this here and oh yeah he mentioned in class that we might need the Fundamental Theorem of Algebra so I'll throw that in there. Oh wait there is a gap here. Lessee.. Assuming this earlier in the arguement fixes that gap. Basta! QED Now onto number 3..