- 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 literaturemathematical 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 allsubset s of "X"), and whosecodomain 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 alevel 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 forinverse function . The two coincide only if "f" is abijection ."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 thepowerset as aposet ordered by inclusion, then the image and preimage functions are monotone.**Examples**1. "f": {1,2,3} → {"a,b,c,d"} defined by $f(x)=left\{egin\{matrix\}\; a,\; mbox\{if\; \}x=1\; \backslash \; d,\; mbox\{if\; \}x=2\; \backslash \; c,\; mbox\{if\; \}x=3.\; end\{matrix\}\; ight.$

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":

**R**→**R**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":

**R**^{2}→**R**defined by "f"("x", "y") = "x"^{2}+ "y"^{2}.The "fibres" "f"

^{ −1}({"a"}) areconcentric circles about the origin, the origin, and theempty set , depending on whether "a">0, "a"=0, or "a"<0, respectively.4. If "M" is a

manifold and "π" :"TM"→"M" is the canonicalprojection from thetangent bundle "TM" to "M", then the "fibres" of "π" are thetangent spaces "T"_{"x"}("M") for "x"∈"M". This is also an example of afiber bundle .**Consequences**Given a function "f" : "X" → "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}→ "f"("A"_{1}) ⊆ "f"("A"_{2})

*"B"_{1}⊆ "B"_{2}→ "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, evenuncountably 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 asemilattice 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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