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The Mathematical Proof Thread
#41
RE: The Mathematical Proof Thread
(2nd October 2016, 01:17)robvalue Wrote: Interesting! I'll have a look at those later if my brain starts working.

Just as an aside, we don't allow 1 to be a prime number because we would lose unique factorisation of integers into primes. For example:

6 = 2 * 3 Unique

6 = 2 * 3 * 1 = 2 * 3 * 1 * 1 = ...

Why do I find humor in this?
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#42
RE: The Mathematical Proof Thread
I don't know Tongue Maths is fun! I've always said so Wink

Cool stuff about the binomial! I think I remember working out a similar proof when I was bored at college, many years ago. I can't remember if it was exactly the same.
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#43
RE: The Mathematical Proof Thread
Prove that C(n-1,k-1) + C(n-1,k)= C(n,k)


Proof strategy.




Proof

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#44
RE: The Mathematical Proof Thread
Hello AF members.  When I first made this thread, I had intended it to be one of those threads where people could post endlessly even if there were a long intermission between posts such as this post.  Hence, I want to apologize to the staff if I'm breaking the rules by posting this.  

With that said, I didn't recall seeing many induction proofs here, so I thought I'd post one for any member who is curious about induction, especially for the younger members (high school) who are getting interested in mathematics.  Here it is:

Prove that 1^2+2^2+…+ (n-1)^2 < n^3/3 < 1^2+2^2+…+ n^2 is true for all positive integers n.


Hint: use mathematical induction.

Background on Mathematical Induction




 
Proof 


"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#45
RE: The Mathematical Proof Thread
Consider the function θ: P(ℤ) → P(ℤ) defined as θ(X)=X^c (note ℤ is the set of all integers).  We can also rephrase this as the function θ from the power set of ℤ to the power set of ℤ is defined as the function θ whose input set X produces an output set equal to the complement of X (or X^c).  Now determine if θ  is injective, surjective, and bijective. This problem was retrieved via Hammack, Richard. (2013) Book of Proof, 2nd ed. Virginia commonwealth University: published by Richard Hammack, pg 205.

Definitions and Terms


Injective, surjective, and bijective functions. 




 
The complement




The law of double complements






The Power Set



 
Proof (My attempt)

 
 
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#46
RE: The Mathematical Proof Thread
Heh. I love listening (reading in this case) to smart people talk about smart stuff. Math, techy stuff, physics, you name it. If there was a room full of people having this conversation, I'd probably be sitting in a corner watching intensely and almost excitedly. Carry on.
Angel
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#47
RE: The Mathematical Proof Thread
(1st February 2017, 22:10)Mamacita Wrote: Heh. I love listening (reading in this case) to smart people talk about smart stuff. Math, techy stuff, physics, you name it. If there was a room full of people having this conversation, I'd probably be sitting in a corner watching intensely and almost excitedly. Carry on.
Angel

You're very kind.  IMO, you are an intelligent poster of this forum and there are probably many topics that you know about which would cause me to be in awe of you. 

With that said, IMO, at best, I have gained an appreciation for mathematics and think it is cool; however, I've got a lot to learn and there are many concepts beyond me, which I will need to master and cultivate before I can be worthy of any sort of merit or whateverSmile. Thanks and Live long and prosper.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#48
RE: The Mathematical Proof Thread
Don't stress Mama... Pretty sure he's a Vulcan, like Alex...
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#49
RE: The Mathematical Proof Thread
Here's a quick, and IMO, interesting video on the Fibonacci numbers by Arthur Benjamin



"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#50
RE: The Mathematical Proof Thread
(8th October 2016, 10:05)mh.brewer Wrote:
(2nd October 2016, 01:17)robvalue Wrote: Interesting! I'll have a look at those later if my brain starts working.

Just as an aside, we don't allow 1 to be a prime number because we would lose unique factorisation of integers into primes. For example:

6 = 2 * 3 Unique

6 = 2 * 3 * 1 = 2 * 3 * 1 * 1 = ...

Why do I find humor in this?

I find it humorous because I misread it as "we don't allow 1 to be prime minister".

So used to Rob's deadpan jokes I thought he was making some kind of math...s pun.
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through friends they always cared about and dreams they left behind, in beauty that they added to our days... in words of wisdom we still carry with us and memories that never will be gone... Those we love are never really lost to us - For everywhere their special love lives on.
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