Properties of images:

Suppose  f  is a function and  C  is a subset of the domain of  f.  Then, by definition,

y  is an element of  f ^→ (C)  (the textbook's notation for this set is simply  f(C) )

iff

there exists  x  in  C  such that  y = f(x).

Consequences:

If  x ∈ C,  then  f(x) ∈ f ^→ (C).
If  y ∈ f ^→ (C), then  ∃ c ∈ C such that  y = f(c).

WARNING!	The definition says	IF x ∈ C,  THEN  f(x) ∈ f → (C).
	It does not say	IF  f(x) ∈ f → (C),  THEN  x ∈ C.
		(This is not true “in general”.)

Suppose f : X → Y and D is a subset of Y.  Then, by definition,

x ∈ f ^← (D)  (the textbook's notation for this set is  f ^-1 (D) )

iff

f(x) ∈ D,

(i.e.,

∃ y in D such that y = f(x);  but this is usually not a good way of thinking of pre-image ).

Consequences:

If  x ∈ f ^← (D),  then  f(x) ∈ D.

If  f(x) ∈ D, then  x ∈ f ^← (D).

Last modified Mar 21, 2008 9:40 AM

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