Image (mathematics)


Image (mathematics)

In mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage, successively, as the function's argument.

Definition

If "f" : "X" → "Y" is a function from set "X" to set "Y" and "x" is a member of "X", then "f"("x"), the image of "x" under "f", is a unique member of "Y" that "f" associates with "x". The image under "f" of the entire domain "X" is often called the range of "f", and is a subset of the codomain "Y".

The image of a subset "A" ⊆ "X" under "f" is the subset of "Y" defined by

:"f" ["A"] = {"y" ∈ "Y" | "y" = "f"("x") for some "x" ∈ "A"}.

When there is no risk of confusion, "f" ["A"] is simply written as "f"("A"). An alternative notation for "f" ["A"] that is common in the older literature mathematical logic and still preferred by some set theorists, is "f" "A".

Given this definition, the image of "f" becomes a function whose domain is the power set of "X" (the set of all subsets of "X"), and whose codomain is the power set of "Y". The same notation can denote either the function "f" or its image. This convention is a common one; the intended meaning must be inferred from the context.

The preimage or inverse image of a set "B" ⊆ "Y" under "f" is the subset of "X" defined by

:"f" −1 ["B"] = {"x" ∈ "X" | "f"("x") ∈ "B"}.

The inverse image of a singleton, "f" −1 [{"y"}] , is a fiber (also spelled fibre) or a level set.

Again, if there is no risk of confusion, we may denote "f" −1 ["B"] by "f" −1("B"), and think of "f" −1 as a function from the power set of "Y" to the power set of "X". The notation "f" −1 should not be confused with that for inverse function. The two coincide only if "f" is a bijection.

"f" can also be seen as a family of sets indexed by "Y", which leads to the notion of a fibred category.

Uniform arrow notations

The traditional notations used in the previous section can be confusing. An alternative [Blyth 2005, p. 5] is to explicitly write the image and preimage as two functions in their own right: f^ ightarrow:mathcal{P}(X) ightarrowmathcal{P}(Y) with f^ ightarrow(A) = { f(a);|; a in A} and f^leftarrow:mathcal{P}(Y) ightarrowmathcal{P}(X) with f^leftarrow(B) = { a in X ;|; f(a) in B}. If we consider the powerset as a poset ordered by inclusion, then the image and preimage functions are monotone.

Examples

1. "f": {1,2,3} → {"a,b,c,d"} defined by f(x)=left{egin{matrix} a, & mbox{if }x=1 \ d, & mbox{if }x=2 \ c, & mbox{if }x=3. end{matrix} ight.

The "image" of {2,3} under "f" is "f"({2,3}) = {"d,c"}, and the "range" of "f" is {"a,d,c"}. The "preimage" of {"a,c"} is "f" −1({"a,c"}) = {1,3}.

2. "f": RR defined by "f"("x") = "x"2.

The "image" of {-2,3} under "f" is "f"({-2,3}) = {4,9}, and the "range" of "f" is R+. The "preimage" of {4,9} under "f" is "f" −1({4,9}) = {-3,-2,2,3}.

3. "f": R2R defined by "f"("x", "y") = "x"2 + "y"2.

The "fibres" "f" −1({"a"}) are concentric circles about the origin, the origin, and the empty set, depending on whether "a">0, "a"=0, or "a"<0, respectively.

4. If "M" is a manifold and "&pi;" :"TM"&rarr;"M" is the canonical projection from the tangent bundle "TM" to "M", then the "fibres" of "&pi;" are the tangent spaces "T""x"("M") for "x"&isin;"M". This is also an example of a fiber bundle.

Consequences

Given a function "f" : "X" &rarr; "Y", for all subsets "A", "A"1, and "A"2 of "X" and all subsets "B", "B"1, and "B"2 of "Y" we have:

*"f"("A"1 ∪ "A"2) = "f"("A"1) ∪ "f"("A"2)
*"f"("A"1 ∩ "A"2) ⊆ "f"("A"1) ∩ "f"("A"2)
*"f" −1("B"1 ∪ "B"2) = "f" −1("B"1) ∪ "f" −1("B"2)
*"f" −1("B"1 ∩ "B"2) = "f" −1("B"1) ∩ "f" −1("B"2)
*"f"("f" −1("B")) ⊆ "B"
*"f" −1("f"("A")) ⊇ "A"
*"A"1 ⊆ "A"2 &rarr; "f"("A"1) ⊆ "f"("A"2)
*"B"1 ⊆ "B"2 &rarr; "f" −1("B"1) ⊆ "f" −1("B"2)
*"f" −1("B"C) = ("f" −1("B"))C
*("f" |"A")−1("B") = "A" ∩ "f" −1("B").

The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:
*fleft(igcup_{sin S}A_s ight) = igcup_{sin S} f(A_s)
*fleft(igcap_{sin S}A_s ight) subseteq igcap_{sin S} f(A_s)
*f^{-1}left(igcup_{sin S}A_s ight) = igcup_{sin S} f^{-1}(A_s)
*f^{-1}left(igcap_{sin S}A_s ight) = igcap_{sin S} f^{-1}(A_s)(here "S" can be infinite, even uncountably infinite.)

With respect to the algebra of subsets, by the above we see that the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism (it does not always preserve intersections).

ee also

*range (mathematics)
*domain (mathematics)
*bijection, injection and surjection
*kernel of a function
*image (category theory)
*preimage attack (cryptography)

Notes

References

*Citation |authorlink=Michael Artin | last= Artin | first= Michael | title= Algebra | edition=| year=1991 | publisher=Prentice Hall| isbn= 81-203-0871-9
* T.S. Blyth, "Lattices and Ordered Algebraic Structures", Springer, 2005, ISBN 1-85233-905-5.


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