Let and . Then . If for then thus since is injective, and thus, is injective.
The other problems is similar.
(i) remember what it means to be injective
we say a function is injective if and only if implies
Proof:
Assume to the contrary that is injective but is not.
since is injective, , and since is not injective, we have and .
but
so as well
but that means we would have and . 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)
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 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.
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
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
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)
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.
Eventhough they are called philosophers over here, these are actually mathematicians. These type of philosophers actually know what they are talking about. For some reason mathematicians that do work in the foundations of math, set theory, logic, category theory, ... are sometimes called philosophers.Originally Posted by Jhevon
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.