Map Well-Defined: Proving Injectivity of a Map

[fk378]

This is a general question...

What is the difference between showing that a map is well-defined and that it is injective?

To prove both can't you show that, given a map x, and elements a,b
if x(a)=x(b) we want to show a=b.

 

[Unassuming]

I think that f(x)=x^2 is well defined but not injective (1-1). I was under the impression that well defined just meant that it is "well-defined" where the domain values are assigned.

f(x)= a large number, that function is not really well-defined.

 

[enigmahunter]

An injective map implies a well-defined map, but a well-defined map does not necessarily imply an injective map.

$$f:X \longrightarrow Y$$
$$a,b \in X$$ and $$f(a), f(b) \in Y$$

For a well-defined map,
a=b implies f(a)=f(b).
(if "a=b implies $$f(a) \neq f(b)$$", then f is not a function ).

For an injective map,
f(a)=f(b) implies a=b.
(You can consider this as a contrapositive way. If a and b are different, then f(a) and f(b) should be different for a map to be injective )