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Well-defined map

  1. Dec 16, 2008 #1
    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.
  2. jcsd
  3. Dec 16, 2008 #2
    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.
  4. Dec 16, 2008 #3
    An injective map implies a well-defined map, but a well-defined map does not necessarily imply an injective map.

    [tex]f:X \longrightarrow Y [/tex]
    [tex]a,b \in X[/tex] and [tex] f(a), f(b) \in Y[/tex]

    For a well-defined map,
    a=b implies f(a)=f(b).
    (if "a=b implies [tex]f(a) \neq f(b)[/tex]", 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 )
    Last edited: Dec 16, 2008
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