Some questions on axiom of choice and zorn's lemma.

[MathematicalPhysicist]

1)axiom of choice: prove that for every set X and for every f:X->X there exists g:X->X such that fogof=f.
2)zorn's lemma: let R be a partial order on X (X a set), prove that there exists a linear order S on X such that R is a subset of S.

well for the second question i used zorn's lemma to find a maximal partial order S. now we assume that (x,y) and (y,x) aren't in S, so i need to prove that T=SU{(a,b)|(a,x),(y,b) in S} is a partial order and thus get a contradiction.
now my problem is to prove that it's a partial order obviously (a,a) is in T cause it's in S, but if let's say for antisymmety, if (a,b) and (b,a) in T, then if both of them in S then obvoiously b=a, but what about the other cases? if (a,b) in S and (b,a) not in S, then (b,x) and (y,a) in S, so (y,b) in S and so is (a,x), but then if (b,x) and (y,b) then (x,y) is in S, which is a contradiction this is why the only possible outcome if for (a,b) and (b,a) to be in S.
is this correct?
and after that obviously i get a contradiction for the maximality of S.
(only need to prove transtivity.

now for the first question, I am given a hint to use an equivalent statement for the axiom of choice that if A is a class of non empty sets A={C_i|i in I}, then there exists a function f:A->U(i in I)C_i such that f(C_i) is in C_i for every i in I, but i don't see how to apply it in here, obviously i need to show that g(f(X))=X, but how?

 

[mmwave]

For the first question, you can use the hint given to show that for every x in X, there exists y in X such that f(y) = x. Then, define g(x) = y for each x in X. This way, g(f(x)) = g(y) = x for every x in X, proving that fogof = f.

For the second question, your approach seems correct. To prove transitivity, you can assume that (a,b) and (b,c) are in T, and then use the definition of T to show that (a,c) must also be in T. This will complete the proof that T is a partial order and lead to a contradiction for the maximality of S.

 

[tacman]

For the first question, you can use the equivalent statement for the axiom of choice to show that there exists a function h:X->X such that h(X) is a singleton set. Then, define g:X->X as g(x)=h(x). This will satisfy the condition fogof=f, since f and g are both functions from X to X.

For the second question, your approach seems correct. To prove transitivity, you can assume that (a,b) and (b,c) are in T, and then use the fact that (a,x) and (y,c) are in S to show that (a,c) is also in S. This will prove that T is a partial order and lead to a contradiction, as you mentioned.

For the first question, you can try to construct a function g:X->X using the given function f:X->X. Consider the set A={C_i|i in I}, where each C_i is the set of all functions from X to X that satisfy fogof=f. Then, using the equivalent statement for the axiom of choice, there exists a function h:A->U(i in I)C_i such that h(C_i) is in C_i for every i in I. This h will give you a way to construct the function g:X->X.