Let and . Then . If for then thus since is injective, and thus, is injective.
The other problems is similar.
we say a function is injective if and only if implies
Assume to the contrary that is injective but is not.
since is injective, , and since is not injective, we have and .
so as well
but that means we would have and . A contradiction
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)
*)In particular his aesthetics.
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
Only a very small school of philosophers reject the rule of excluded middle.
Law of excluded middle - Wikipedia, the free encyclopedia
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
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.