Problem: find the solution set of x^2 + 1 < 0.
Burn, Robert wrote:
> I am not sure that any of us have (in these mailings) grappled with the concept of the empty set.
> I believe Ed Dubinsky's report that grammatical practice can make orthodox notation habitual, and then, in retrospect, the concept is accomodated.
> Is there a problem, at undergraduate level, to which the concept of the empty set is a satisfying solution?
> I notice that calculus/analysis in the 19th century managed without it.
> Johnston's example is significant in that it gives a neat formula for the cardinality of a power set.
>
> Bob Burn
>
>
>
>
> -----Original Message-----
> From: Post-calculus mathematics education on behalf of Kazimierz Wiesak
> Sent: Sat 27/10/2007 23:06
> To: [log in to unmask]
> Subject: Re: Set notation
>
>
> There seems to be two problems: the conceptual problem and the notational problem.
> Confusion of notation doesn't necessarily mean confusion of concepts.
> Example: list all subsets of {1,2,3}.
> {O, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
> Notice inconsistency of notation, all subsets are in brackets except one O.
> Students can be tempted to get "notational consistency" and put O in brackets.
>
> We should admit that {O} is a notational possibility - albeit potentially confusing - for "the set consisting of nothing". This possibility may explain notations used by students. Such a language does not necessarily mean that the student is not aware of distinction between "empty set" and "the set consisting of the empty set only". It may mean that his "sense of language" hasn't accepted yet the standard notation used by mathematicians who, to avoid confusion, chose O for empty set instead of {O}.
>
> Kazimierz
>
> At 06:48 AM 10/26/07, Anderson Johnston wrote:
>
>
>
>
> The context wasn't really to do with sets per se, but I remember a tutorial session
> in which enlightenment struck some of the students when, looking at the probabilities
> of choosing a certain number of objects from a given collection, it dawned on them
> that one legitimate operation was to choose none of the objects (and that the
> likelihood of that was the same as choosing all of them).
>
> Johnston
>
> PS. Good to see you are well and active, Bob!
>
>
>
> ________________________________
>
> From: Post-calculus mathematics education on behalf of Burn, Robert
> Sent: Fri 19/10/2007 21:39
> To: [log in to unmask]
> Subject: Re: Set notation
>
> Ed Dubinsky cannot get his messages read by MATHEDU, so here is the response he wanted to make.
>
> Bob Burn
> Research Fellow, Exeter University
> Sunnyside
> Barrack Road
> Exeter EX2 6AB
> 01392-430028
>
>
>
> -----Original Message-----
> From: Ed Dubinsky [mailto:[log in to unmask]]
> Sent: Wed 17/10/2007 13:06
> To: Burn, Robert
> Cc: [log in to unmask]
> Subject: Re: Set notation
>
> See below for responses.
>
> On Wed, 17 Oct 2007, Burn, Robert wrote:
>
> > Just two thoughts.
> > 1. To regard the empty set as a thing, is quite a step. I dont think it
> > rates as a thing from the perspective of Euclid's Elements.
>
> I agree completely.
>
> > 2. Some autobiography might be illuminating: at what point did readers
> > of this list recognise the distinction between the empty set and {the
> > empty set}? I think for me it was at the construction of the natural
> numbers.
>
> I don't know about myself, but I can tell you what works really well
> (that means, a high percentage of students get it): Having students write
> computer programs that construct sets (including the empty set) and
> perform actions on them such as checking their cardinality, forming
> unions, intersections, etc.
>
> Ed
>
>
> > Bob Burn
> > Research Fellow, Exeter University
> > Sunnyside
> > Barrack Road
> > Exeter EX2 6AB
> > 01392-430028
> >
> >
> >
> > -----Original Message-----
> > From: Post-calculus mathematics education on behalf of Smith, Alexander J.
> > Sent: Sun 07/10/2007 23:50
> > To: [log in to unmask]
> > Subject: Re: Set notation
> >
> > Let us not forget the following word of Feynman.
> >
> > (My humble experience is that it is a happy event when an undergraduate mathematics major can intuitively distinguish between the empty set and the set which contains only the empty set.)
> >
> > Feynman's words:
> >
> > The power of instruction is seldom of much efficacy except in
> > those happy dispositions where it is almost superfluous.
> >
> > There isn't any solution to this problem of education other than
> > to realize that the best teaching can be done only when there
> > is a direct individual relationship between a student and a good
> > teacher--a situation in which the student discusses the ideas,
> > thinks about the things, and talks about the things. It's impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned.
> >
> >
> > ___________________________________
> > From: Post-calculus mathematics education [[log in to unmask]] On Behalf Of Ralph A. Raimi [[log in to unmask]]
> > Sent: Sunday, October 07, 2007 5:17 PM
> > To: [log in to unmask]
> > Subject: Re: Set notation
> >
> > On Sun, 7 Oct 2007, Murray Eisenberg wrote:
> >
> >> I just experienced this phenomenon (again!) in the first exam in our proofs
> >> course, where the question was to list the elements of the power set of
> >> {1,2,3}.
> >>
> >> Several students gave the answer as
> >>
> >> {Ø}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
> >>
> >> or as ... but to wonder what theory can overcome general linguistic
> >> insensitivity. The relevant research might involve much earlier stages of
> >> mental and linguistic development.
> >
> > An anecdote from the days of "the new math" in America, when
> > elementary school teachers were instructed to tell the kiddies about sets,
> > unions and such:
> >
> > The teacher, having seemingly absorbed the idea of distinguishing
> > between a set and its members, and bent on transferring the lesson to her
> > class, asks "the set of all boys to stand up", and then, "the set of all
> > girls to stand up".
> >
> > (Excuse me: I meant "bent on transferring the lesson to the
> > members of her class". The class cannot absorb a lesson any more than
> > the set of all boys can stand up.)
> >
> > Which is to say that we (even mathematicians) are accustomed
> > to conflating the set with its members in daily speech, and have really no
> > reason to be pedantic about it until careful reasoning in mathematics
> > requires it of us. I see little reason to try to teach such things before
> > university mathematics begins to consider theorems regarding which, and
> > regarding whose proofs, this distinction has some application. As we all
> > have seen, the lesson simply won't go over, except for some few who don't
> > need it anyhow, not even if they aspire to careers in science or
> > mathematics, for they will learn it easily enough when the time comes.
> >
> > Ralph A. Raimi Tel. 585 275 4429 or (home) 585 244 9368
> > Dept. of Mathematics, Univ.of Rochester, Rochester, NY 14627
> > < http://www.math.rochester.edu/people/faculty/rarm/ <http://www.math.rochester.edu/people/faculty/rarm/> >
> >
> > "Algebra is conducive to symbolic reasoning." ....PSSM, p.345
> >
> >
>
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> Ed Dubinsky
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