- #1
Vadermort
- 11
- 0
Can someone pretty please explain to me the axiom of choice as unrigorously and casually as possible ;P.
And then you could include a rigorous explanation if you wish, but I mean I could just go to wikipedia for that..which I already did... in fact I think I'll include the wiki explanation and parts I have trouble with...
FROM WIKIPEDIA:
First statement of the axiom (the very beyond me explanation):
for every family (Si)i ∈ I of nonempty sets there exists a family (xi)i ∈ I of elements with xi ∈ Si for every i ∈ I.
1. ∈ <---what is this?
2. What about the rest of it? (really just don't have a firm grasp of the notation here, the only domains of math I have experience with are middle school algebra and calculus)
Second statement (the slightly less beyond me explanation):
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
For any set X of nonempty sets, there exists a choice function f defined on X.
1.) Is a collection of sets the same thing as a set of sets? or are sets defined by certain qualities e.g the natural numbers constitute a set...
2.) my attempt at restating it: for every set s in X, f(X) will yield an f(s) ...and why is this important?
And then you could include a rigorous explanation if you wish, but I mean I could just go to wikipedia for that..which I already did... in fact I think I'll include the wiki explanation and parts I have trouble with...
FROM WIKIPEDIA:
First statement of the axiom (the very beyond me explanation):
for every family (Si)i ∈ I of nonempty sets there exists a family (xi)i ∈ I of elements with xi ∈ Si for every i ∈ I.
1. ∈ <---what is this?
2. What about the rest of it? (really just don't have a firm grasp of the notation here, the only domains of math I have experience with are middle school algebra and calculus)
Second statement (the slightly less beyond me explanation):
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
For any set X of nonempty sets, there exists a choice function f defined on X.
1.) Is a collection of sets the same thing as a set of sets? or are sets defined by certain qualities e.g the natural numbers constitute a set...
2.) my attempt at restating it: for every set s in X, f(X) will yield an f(s) ...and why is this important?