http://common.replayers.com/temp/maths2.JPG

This is the second part of two questions that I could really do with some help with.

Any help with any part would be super!

Many Thanks in advance,

The Maths Dude

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- Jan 15th 2008, 02:19 PMTheMathsDudeProof Question 2
http://common.replayers.com/temp/maths2.JPG

This is the second part of two questions that I could really do with some help with.

Any help with any part would be super!

Many Thanks in advance,

The Maths Dude - Jan 15th 2008, 02:49 PMThePerfectHacker
Let $\displaystyle f: A\mapsto B$ and $\displaystyle g: B\mapsto C$. Then $\displaystyle g\circ f : A\mapsto C$. If $\displaystyle f(x)=f(y)$ for $\displaystyle x,y\in A$ then $\displaystyle g\circ f(x) = g\circ f(y)$ thus $\displaystyle x=y$ since $\displaystyle g\circ f$ is injective, and thus, $\displaystyle f$ is injective.

The other problems is similar. - Jan 15th 2008, 03:08 PMJhevon
(i) remember what it means to be injective

we say a function $\displaystyle f(x)$ is injective if and only if $\displaystyle f(a) = f(b)$ implies $\displaystyle a = b$

**Proof:**

Assume to the contrary that $\displaystyle (g \circ f)(x)$ is injective but $\displaystyle f(x)$ is not.

since $\displaystyle (g \circ f)(x)$ is injective, $\displaystyle (g \circ f)(a) = (g \circ f)(b) \implies a = b$, and since $\displaystyle f$ is not injective, we have $\displaystyle f(a) = f(b)$ and $\displaystyle a \ne b$.

but $\displaystyle (g \circ f)(x) = g(f(x))$

so $\displaystyle g(f(a)) = g(f(b)) \implies f(a) = f(b)$ as well

but that means we would have $\displaystyle f(a) = f(b)$ and $\displaystyle a = b$. A contradiction

QED

try to manipulate the definition of surjective to prove the others. you can find the definition here

EDIT: This is very similar to TPH's post. except he used a direct proof where I used contradiction. (direct proofs are generally preferred) - Jan 15th 2008, 03:14 PMThePerfectHacker
- Jan 15th 2008, 03:18 PMJhevon
- Jan 15th 2008, 03:24 PMThePerfectHacker
I assume that has to do with one's philosophy on math*. There are some mathematicians that hate with a love indirect proof (though they do not deny them, just hate them). Also, there are some mathematicians that favor and existence proof over a constructive proof. And some mathematicians are even more extreme and perfer finitism, like the famous algebraist (Bow) Leopold Kronecker. I myself really do not bother myself with these things, I like existence proofs without any constructions, because all of these approaches work.

*)In particular his aesthetics. - Jan 15th 2008, 03:44 PMJhevon
like you, i don't really like to bother wiuth these things. aesthetics is important to me to some extent, but not to the extent that i would write of a method of doing something or hate it with a passion. if it is proven to work, i'm fine with it. however, i don't know the poster's professor's outlook. he might be someone who hates using indirect proofs when a direct proof can be done, so i say, go with what's safe.

Aside: one of the things i like about math is that it doesn't matter what your professor's personal opinion is, if it's right and mathematically sound, he has to mark it right, regardless of his biases. this is not so with the arts, and even some of the sciences. but anyway, getting good grades in college is half about pleasing your professor, might as well practice on our math professors as well - Jan 15th 2008, 04:07 PMPlato
That my friend is just B.S.

Only a very small school of philosophers reject the rule of*excluded middle*.

Law of excluded middle - Wikipedia, the free encyclopedia - Jan 15th 2008, 04:11 PMJhevon
- Jan 15th 2008, 04:41 PMtopsquark
- Jan 15th 2008, 04:54 PMJhevon
the ideas used were exactly the same, but the structure or presentation was different. things like "assume to the contrary" or "this leads to a contradiction" makes it seem unaesthetic to some people. proving something by

*contradiction*seems unsavory to them somehow. it's like arguing for something right based on something wrong (that's not how i interpret it, but i assume they would) - Jan 15th 2008, 05:30 PMThePerfectHacker
I agree with Jhevon an indirect proof is really a direct proof by the contrapositive. But for some reason it loses its elegance.

Maybe a bad analogy: Playing the 5th with an orchestra is much more elegant then listening it to be played by a band using synthensized instruments.

Quote:

Originally Posted by**Jhevon**

- Jan 15th 2008, 05:59 PMPlato
Little you know about this, my friend!

I dare say that I am the only member of the board who has been a NHF fellow in the philosophy of mathematics. There is a**very**small school of so-called*intuitionalist*who reject the law of*excluded middle*. This notion has been active in the ‘mathematics education’ community. It has produced a plethora of doctoral thesis in*mathematics education*.

Now this may by strictly my owe biases, but I personally do not know any active PhD level mathematician who accepts that position. The truth is: there are just so many other**real problems in mathematics education**it is lunacy go against a majority of mathematicians. - Jan 16th 2008, 01:10 AMJhevon
we're starting to go so much off topic that we might have to move this thread to the "Philosophy of Math" forum :D it's a good talk though

did you get through the other problems ok, TheMathsDude? - Jan 16th 2008, 01:13 AMTheMathsDude
Finding it hard to manipulate the surjective definition to fit the previous examples written.

Any clues?