Show that the function(s) is well defined

[tylerc1991]

Homework Statement

Show that the functions are well defined:
(a) f: Q -> Z defined by f(a/b) = a
(b) f: Q -> Q defined by f(a/b) = a^2 / b^2

Homework Equations

Q - rationals
Z - integers

The Attempt at a Solution

(a) pretty obvious, as I can come up with an example to show that f is not well defined. f(1/2) = 1, but f(2/4) = 2.

(b) since ca/cb = a/b for some constant c, f(a/b) = f(ca/cb) = (c^2*a^2)/(c^2*b^2) = a^2 / b^2 => f is well defined.

Not so sure about (b) here. In fact, I'm not even certain about the definition of a 'well-defined function'. Seeing as how my book does a poor job could someone give me a little help. Thank you!

 

[Gib Z]

I think they had an implicit assumption that when you write a rational as a/b, it is already in its reduced form. Then the functions are well defined like the question says they should be.

A well defined function has no special meaning- it's just a function that is defined "well", in that it's not ambiguous, and it works like a normal function - so it should have precisely 1 output value for each input value, you should be able to input any value from the domain, and the outputs should be in the given codomain. So after you do those two questions, try these:

Why are these NOT well defined functions?
(a) f: R -> Z defined by f(a/b) = a
(b) f: Q -> Q defined by f(a/b) = (a/b)^2 + e