Axiom of choice
I am given two forms of the axiom of choice, and I need to show that they are equivalent. Can someone help?
Form 1: a relation a function and and
Form 2: [If , y is not equal the empty set, then is not equal the empty set.
Where is the Cartesian product defined as the set of functions f with dom f=X and for all y in X, f(y) is in y.
Form 1 says that "For any relation R, there is a function with ".
Originally Posted by namelessguy
We define a relation R such that , where a choice function C is defined as such that is an element of . Since and is not empty, the existence of choice function C is followed from a Form1, satisfying the Form 2.
Conversely, assume Form 2.
Let ; let .
Now, we have and for each i, .
That means, there exists a choice function such that we retrieve for each .
Now, consider any relation R. The existence of a choice function C provides us with a mapping from to .
Then, , i.e., . Thus, .