Functions:Forward Image - Revision history

Khanh at 21:11, 6 November 2021

2021-11-06T21:11:21Z

Lila: /* Resources */

2021-10-12T16:09:47Z

Resources

Lila: /* Resources */

2021-10-12T16:09:28Z

Resources

Lila: /* Inverse image of a function */

2021-10-12T16:08:17Z

Inverse image of a function

Lila: /* Inverse image of a function */

2021-10-12T16:07:32Z

Inverse image of a function

Lila at 16:05, 12 October 2021

2021-10-12T16:05:45Z

Lila at 16:03, 12 October 2021

2021-10-12T16:03:07Z

Lila: /* Definition */

2021-10-12T16:02:19Z

Definition

Lila: /* Resources */

2021-10-12T15:59:38Z

Resources

Lila at 15:58, 12 October 2021

2021-10-12T15:58:48Z

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 With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections).
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia
+Content obtained and/or adapted from:
+* [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics), Wikipedia] under a CC BY-SA license

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 ==Resources==
 * [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia
 ===References===

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 ==Resources==
 * [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia
 ===References===

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 The inverse image of a singleton set, denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}[y],</math> is also called the fiber or fiber over <math>y</math> or the level set of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>
-For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for [[inverse function]], although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>
+For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>
 ==Examples==

@@ Line 27: / Line 27: @@
 ==Inverse image of a function==
-Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by
+Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by <math>f^{-1}[ B ] = \{ x \in X \,|\, f(x) \in B \}.</math>
 Other notations include <math>f^{-1}(B)</math> and <math>f^{-}(B).</math>

@@ Line 25: / Line 25: @@
 ===Generalization to binary relations===
 If <math>R</math> is an arbitrary binary relation on <math>X\times Y</math>, then the set <math>\{y\in Y:xRy \text{ for some } x\in X\}</math> is called the image, or the range, of <math>R</math>. Dually, the set <math>\{x\in X:xRy{\text{ for some }}y\in Y\}</math> is called the domain of <math>R</math>.
 ==Examples==

← Older revision		Revision as of 16:03, 12 October 2021
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	Given <math>y</math>, the function <math>f</math> is said to "take the value <math>y</math>" or "take <math>y</math> as a value" if there exists some <math>x</math> in the function's domain such that <math>f(x)=y</math>. Similarly, given a set <math>S</math>, <math>f</math> is said to "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain.		Given <math>y</math>, the function <math>f</math> is said to "take the value <math>y</math>" or "take <math>y</math> as a value" if there exists some <math>x</math> in the function's domain such that <math>f(x)=y</math>. Similarly, given a set <math>S</math>, <math>f</math> is said to "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain.
−
−	~~==Inverse image==~~
−
−	Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by
−	~~<math display=block>f^{-1}[ B ] = \{ x \in X \,\|\, f(x) \in B \}.</math>~~
−
−	Other notations include <math>f^{-1}(B)</math><ref>{{Cite web\|date=2020-03-25\|title=Comprehensive List of Algebra Symbols\|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/\|access-date=2020-08-28\|website=Math Vault\|language=en-US}}</ref> and <math>f^{-}(B).</math>{{sfn\|Dolecki\|Mynard\|2016\|pp=4-5}}
−	The inverse image of a [[Singleton (mathematics)\|singleton set]], denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}[y],</math> is also called the [[Fiber (mathematics)\|fiber]] or fiber over <math>y</math> or the [[level set]] of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>
−
−	For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for [[inverse function]], although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>

	===Image of a subset===		===Image of a subset===

← Older revision		Revision as of 16:02, 12 October 2021
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	Given <math>y</math>, the function <math>f</math> is said to "take the value <math>y</math>" or "take <math>y</math> as a value" if there exists some <math>x</math> in the function's domain such that <math>f(x)=y</math>. Similarly, given a set <math>S</math>, <math>f</math> is said to "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain.		Given <math>y</math>, the function <math>f</math> is said to "take the value <math>y</math>" or "take <math>y</math> as a value" if there exists some <math>x</math> in the function's domain such that <math>f(x)=y</math>. Similarly, given a set <math>S</math>, <math>f</math> is said to "take a value in <math>S</math>" if there exists some <math>x</math> in the function's domain such that <math>f(x)\in S</math>. However, "<math>f</math> takes all values in <math>S</math>" and "<math>f</math> is valued in <math>S</math>" means that <math>f(x)\in S</math> for every point <math>x</math> in <math>f</math>'s domain.
		+
		+	==Inverse image==
		+
		+	Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by
		+	<math display=block>f^{-1}[ B ] = \{ x \in X \,\|\, f(x) \in B \}.</math>
		+
		+	Other notations include <math>f^{-1}(B)</math><ref>{{Cite web\|date=2020-03-25\|title=Comprehensive List of Algebra Symbols\|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/\|access-date=2020-08-28\|website=Math Vault\|language=en-US}}</ref> and <math>f^{-}(B).</math>{{sfn\|Dolecki\|Mynard\|2016\|pp=4-5}}
		+	The inverse image of a [[Singleton (mathematics)\|singleton set]], denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}[y],</math> is also called the [[Fiber (mathematics)\|fiber]] or fiber over <math>y</math> or the [[level set]] of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>
		+
		+	For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for [[inverse function]], although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>

	===Image of a subset===		===Image of a subset===

← Older revision		Revision as of 15:59, 12 October 2021
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	==Resources==		==Resources==
	* [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia		* [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia
		+
		+	===References===
		+	* Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9.
		+	* Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5..
		+	* Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
		+	* Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403.
		+	* Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1.
		+	* Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.

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 |}
-The results relating images and preimages to the ([[Boolean algebra (structure)|Boolean]]) algebra of [[Intersection (set theory)|intersection]] and [[Union (set theory)|union]] work for any collection of subsets, not just for pairs of subsets:
+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:
 * <math>f\left(\bigcup_{s\in S}A_s\right) = \bigcup_{s\in S} f\left(A_s\right)</math>
@@ Line 138: / Line 138: @@
 * <math>f^{-1}\left(\bigcup_{s\in S}B_s\right) = \bigcup_{s\in S} f^{-1}\left(B_s\right)</math>
 * <math>f^{-1}\left(\bigcap_{s\in S}B_s\right) = \bigcap_{s\in S} f^{-1}\left(B_s\right)</math>
-(Here, <math>S</math> can be infinite, even [[uncountably infinite]].)
+(Here, <math>S</math> can be infinite, even uncountably infinite.)
-With respect to the algebra of subsets described above, the inverse image function is a [[lattice homomorphism]], while the image function is only a [[semilattice]] homomorphism (that is, it does not always preserve intersections).
+With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections).
 ==Resources==
 * [https://en.wikipedia.org/wiki/Image_(mathematics) Image (mathematics)], Wikipedia