# Operators

43 messages
123
Open this post in threaded view
|

## Operators

 OK, I know this is a stupid question so please go easy. What I would like to know is if the differentiation operator is usually the first operator that students encounter, or is there one (or more) they are already familiar with? Thanks. - -Timotha ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 The inverse of a function is an operator. Bob Hansen On 5/22/15, 9:59 AM, "Timotha Trigg" <[hidden email]> wrote: OK, I know this is a stupid question so please go easy. What I would like to know is if the differentiation operator is usually the first operator that students encounter, or is there one (or more) they are already familiar with? Thanks. - -Timotha ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg On Fri, 22 May 2015 07:59:59 -0600, Timotha Trigg <[hidden email]>   wrote: > OK, I know this is a stupid question so please go easy. What I would   > like to know is if the differentiation operator is usually the first   > operator that students encounter, or is there one (or more) they are   > already familiar with? > > Thanks. > > - -Timotha > > ------- End of Forwarded Message > If you mean the first operator that they are encouraged to think of as   something that transforms functions into other functions, it probably is.   But it isn't the first such that they encounter.  For just one example,   there are the translation operators, which transform the function f: x -->   f)x) into the function f_h: x --> f(x - h).  While they encounter such   operators, they're rarely asked to think of them as "operators". Given the difficulty the notion of "operator" seems to cause even calculus   students, that's probably wise.  Part of the trouble probably arises from   our premature (in the sense of placement in curriculum) formalization of   real-valued functions of a single real variable as being certain sets of   ordered pairs of numbers. - --Louis A. Talman    Department of Mathematical and Computer Sciences    Metropolitan State University of Denver     ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

Open this post in threaded view
|

## Re: Operators

 In reply to this post by Robert Hansen On Fri, May 22, 2015 at 8:49 AM, Robert Hansen wrote: The inverse of a function is an operator. Bob HansenHere you get the standard sage and pious nonsense from our sage-in-residence.  Every list should have at least one of these clowns.Kirby
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Louis Talman Given the difficulty the notion of "operator" seems to cause even calculus students, that's probably wise.  Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of real-valued functions of a single real variable as being certain sets of ordered pairs of numbers. - --Louis A. Talman   Department of Mathematical and Computer Sciences   Metropolitan State University of DenverHere I'd say Lou has already bought into a specific namespace, as known to him as water to a fish.  He's a native here.But in Alien Math, we say:   . A = 10is using "the assignment operator".  + - / * are the "four operators" used in typical arithmetic.  Another operator, %, is usually not taught in Lower48 which is another reason the colleges are mostly what we'd call "remediation academies" in translation.Typical REPL (dialog with an interpreter -- "like Eliza" says Wayne):   . 150 % 750   . 48394579717912 % 4884208Kirby
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg Lou Talman says: >Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of real-valued functions of a single real variable as being certain sets of ordered pairs of numbers. How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it  say, from the perspective you would prefer, that's just a guess. Cheers, Joe N ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by kirby urner-4 On 5/22/15, 2:07 PM, "kirby urner" <[hidden email]> wrote: On Fri, May 22, 2015 at 8:49 AM, Robert Hansen  wrote: The inverse of a function is an operator. Bob Hansen Here you get the standard sage and pious nonsense from our sage-in-residence.   Every list should have at least one of these clowns. What’s wrong? I didn’t use python? Bob Hansen
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg Robert Hansen wrote: http://mathforum.org/kb/message.jspa?messageID=9779516> The inverse of a function is an operator. You might be thinking of "relation". The typical usage of the term "operator" is that it's just a function, except that its domain (and often its range as well) are a little more exotic than the real numbers (or n-dimensional Euclidean space). Thus, there are operators defined on matrices (e.g. the adjoint operator, the exponential operator, the transpose operator, etc.), operators defined on appropriate sets of functions (e.g. various differentiation operators, various integration operators, the translation operator that Louis Talman mentioned, etc.), operators defined on subsets of a topological space (the closure operator, the interior operator, the derived set operator, the frontier operator, etc.), and other kinds, but in all cases we're still talking about relations that are functions. That is, each of these operators associates to each object in its domain exactly one object in its range. So why say "operator" instead of "function"? I think it's just convention, which began before the word "function" began to take on the abstract meaning it now has (in higher level math, at least), back when "function" meant the kinds of things you study in calculus (single variable and multi-variable). Incidentally, people even work with operators whose domain and range are various sets of operators. See my comments here: http://math.stackexchange.com/questions/372429/is-there-a-such-thing-as-an-operator-of-operators-in-mathematicsDave L. Renfro ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg Kirby says: >I think you might be implying something similar to what Joe did: that there might be a "right answer" to this question. Talkin' about me? I don't follow. How/when did I do that? Cheers, Joe N ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Joe Niederberger On Fri, 22 May 2015 12:49:29 -0600, Joe Niederberger   <[hidden email]> wrote: > Lou Talman says: >> Part of the trouble probably arises from our premature (in the sense of   >> placement in curriculum) formalization of > real-valued functions of a single real variable as being certain sets of   > ordered pairs of numbers. > > How so? I've heard you say similar things (all directed at functions as   > ordered pairs) but I don't appreciate the nature of the problem you see.   > In fact, I would think the abstraction would actually help, but not   > having learned it  say, from the perspective you would prefer, that's   > just a guess. > > Cheers, > Joe N > > ------- End of Forwarded Message > The issue is unnecessary formalization that tends, for the beginner, to   obscure rather than illuminate.  At what point of, say, the calculus   curriculum, is any use made of the nature of a function as a set of   ordered pairs?  If there is none, then why introduce the definition at   that level or before? - --Louis A. Talman    Department of Mathematical and Computer Sciences    Metropolitan State University of Denver     ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Dave L. Renfro I was thinking of “a function from a function” to distinguish from “a value from a value”.  Bob Hansen On 5/22/15, 4:21 PM, "Dave L. Renfro" <[hidden email]> wrote: You might be thinking of "relation". The typical usage of the term "operator" is that it's just a function, except that its domain (and often its range as well) are a little more exotic than the real numbers (or n-dimensional Euclidean space). Thus, there are operators defined on matrices (e.g. the adjoint operator, the exponential operator, the transpose operator, etc.), operators defined on appropriate sets of functions (e.g. various differentiation operators, various integration operators, the translation operator that Louis Talman mentioned, etc.), operators defined on subsets of a topological space (the closure operator, the interior operator, the derived set operator, the frontier operator, etc.), and other kinds, but in all cases we're still talking about relations that are functions. That is, each of these operators associates to each object in its domain exactly one object in its range. So why say "operator" instead of "function"? I think it's just convention, which began before the word "function" began to take on the abstract meaning it now has (in higher level math, at least), back when "function" meant the kinds of things you study in calculus (single variable and multi-variable). Incidentally, people even work with operators whose domain and range are various sets of operators. See my comments here:
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg Lou Talman says: >The issue is unnecessary formalization that tends, for the beginner, to obscure rather than illuminate. At what point of, say, the calculus curriculum, is any use made of the nature of a function as a set of ordered pairs? If there is none, then why introduce the definition at that level or before? Oh, reconsidering, we're probably not so far apart. My 60s "New Math" showed little cartoons of "function machines" and whatnot, but didn't really delve into formal definitions of ordered pairs. The emphasis was always on the "to-one" requirement, and the fact that one has  "sets" (informal) for the domain and range (or codomain). Never really thought much about the formalities of it all till Halmos' "Naive Set Theory". I liked the way he treaded carefully around the issue of some people insisting their be some kind of "active agent" (not his words) to functions. I know your favorite formulation is to speak of a "rule" that gives one element of the range for each element of the domain; I like the word "associate" better, but its of little import I think. I also like simple diagrams employing single headed arrows. Cheers, Joe N ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Dave L. Renfro On Fri, May 22, 2015 at 1:21 PM, Dave L. Renfro wrote:<< ... >>  So why say "operator" instead of "function"? I think it's just convention, which began before the word "function" began to take on the abstract meaning it now has (in higher level math, at least), back when "function" meant the kinds of things you study in calculus (single variable and multi-variable).Yes, just a convention.Python the math notation (runnable) clearly indicatesthe equivalence of operators with functions in the sense that A + B is equivalently A.__add__(B), i.e.if A, B are members of domains where "+" makes sense between them, then feed one to the other,and if they're of the same type, which method is picked (__add__) will make no difference.Ditto for *, /, - which are __mul__, __truediv__ and __sub__ respectively.  We can also negate.  Thecoder of the type gets to design behavior for eachof these key operators, setting the stage for Abstract Algebra concepts later.That's a segue to another usual next topic:  "binary versus unary" operators, with - being a paradigm example of one that works in both roles, as -3 andas 0-3, which are equal ( -3 == 0-3 ).  But then somewill argue -3 is not an operator in action but thestatic member of integer set i.e. -3 is a name, not a verb.  These fine points dog the beginner.Kirby  Incidentally, people even work with operators whose domain and range are various sets of operators. See my comments here: http://math.stackexchange.com/questions/372429/is-there-a-such-thing-as-an-operator-of-operators-in-mathematics Dave L. Renfro ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Louis Talman I would speculate that the introduction of the notion of function is to distinguish from 'relation' - so that differentiation is manageable at an early stage.Gary TupperOn Saturday, May 23, 2015, Louis Talman <[hidden email]> wrote:On Fri, 22 May 2015 12:49:29 -0600, Joe Niederberger <[hidden email]> wrote: Lou Talman says: Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of real-valued functions of a single real variable as being certain sets of ordered pairs of numbers. How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it  say, from the perspective you would prefer, that's just a guess. Cheers, Joe N ------- End of Forwarded Message The issue is unnecessary formalization that tends, for the beginner, to obscure rather than illuminate.  At what point of, say, the calculus curriculum, is any use made of the nature of a function as a set of ordered pairs?  If there is none, then why introduce the definition at that level or before? - --Louis A. Talman   Department of Mathematical and Computer Sciences   Metropolitan State University of Denver   ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Joe Niederberger On 5/22/15, 2:49 PM, "Joe Niederberger" <[hidden email]> wrote: How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it  say, from the perspective you would prefer, that's just a guess. I’ve never seen a curriculum in high school where abstraction came out strong. I’ve seen abstraction in high school math, but not where it blossomed like it does in upper or graduate level college classes. I’ve always taken that to mean something about cognition and the stages of coming of age in such subjects. Bob Hansen
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Louis Talman At 08:04 AM 5/23/2015, Louis Talman wrote: >On Fri, 22 May 2015 12:49:29 -0600, Joe Niederberger ><[hidden email]> wrote: > >>Lou Talman says: >>>Part of the trouble probably arises from our premature (in the >>>sense of  placement in curriculum) formalization of real-valued >>>functions of a single real variable as being certain sets of >>>ordered pairs of numbers. >> >>How so? I've heard you say similar things (all directed at >>functions as ordered pairs) but I don't appreciate the nature of >>the problem you see.In fact, I would think the abstraction would >>actually help, but not having learned it  say, from the perspective >>you would prefer, that's  just a guess. >The corrected she is unnecessary formalization that tends, for the >beginner, to  obscure rather than illuminate.  At what point of, >say, the calculus curriculum, is any use made of the nature of a >function as a set of ordered pairs?  If there is none, then why >introduce the definition at that level or before? Because the New Math gurus of SMSG so dictated.  Inappropriate formalism where their seminal meeting was held in 1958 at Stanford and Ed Begle did his best to persuade Stanford's universally respected mathematics problem-solving guru, George Polya, to participate in the group.  He refused because he viewed it as a completely misguided enterprise and doomed for appropriate eventual failure in most schools with most students.  As with the "Moore Method" for college math, he was never addressing all math students at the precollegiate level (even at the collegiate level) much less all students in all programs led by poorly prepared teachers with little interest or ability in mathematics.  As with Moore at UT Austin, he was mostly interested in graduate-level students of Stanford quality but extended that interest to students with exceptional mathematics ability and interest at any level.  He recognized full well that being a powerful problem solver requires an extensive "toolbox" of concepts and algorithms.  In attacking a new situation, the more parts that can be seen as a special case of something well-known, the more one's brain can focus attention on genuinely new aspects  Think Newton's famous quote on his ability to see further and that was Newton, not ordinary students with ordinary teachers much less worse than ordinary on both counts. Bringing things up-to-date, Common Core-Math advocates are misusing Polya to support their "discovery" pedagogy although some recognize the inappropriateness: http://truthinamericaneducation.com/common-core-state-standards/ccss-content-issues-reviews/common-core-standards-for-mathematical-practice-part-i/Lone voices crying in the wilderness aside, Santayana is alive and well. Wayne ------- End of Forwarded Message
Open this post in threaded view
|

## Re: Operators

 In reply to this post by gatupper I don't think this is something that necessitates the ordered-pairs definition at the level of calculus or before.On Sat, 23 May 2015 15:06:30 -0600, Gary Tupper <[hidden email]> wrote:I would speculate that the introduction of the notion of function is to distinguish from 'relation' - so that differentiation is manageable at an early stage.Gary TupperOn Saturday, May 23, 2015, Louis Talman <[hidden email]> wrote:On Fri, 22 May 2015 12:49:29 -0600, Joe Niederberger <[hidden email]> wrote: Lou Talman says: Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of real-valued functions of a single real variable as being certain sets of ordered pairs of numbers. How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it  say, from the perspective you would prefer, that's just a guess. Cheers, Joe N ------- End of Forwarded Message The issue is unnecessary formalization that tends, for the beginner, to obscure rather than illuminate.  At what point of, say, the calculus curriculum, is any use made of the nature of a function as a set of ordered pairs?  If there is none, then why introduce the definition at that level or before?  -- --Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver
Open this post in threaded view
|

## Re: Operators

 In reply to this post by Timotha Trigg R Hansen says: >I was thinking of ?a function from a function? to distinguish from ?a value from a value?. For some sets of functions whose inverse relations are functions (any way you want to say that) then indeed you have an "function inversion operator" as long as the domain (invertible functions of some type) is appropriately specified. I strongly doubt that "all invertible functions" would be a proper set. Sounds too big. Anyone? Cheers, Joe N ------- End of Forwarded Message