Re: Review of Mueckenheims book.

MoeBlee wrote:
On Mar 6, 3:07 pm, David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:
The vast majority of math textbooks define functions as maps between two
specified sets.

I don't know about the vast majority. I've seen different kinds of
definitions in different books. Plenty of textbooks in abstract
algebra, analysis, and topology DO give the usual set theoretic
definition of 'function', while other textbooks in those subjects (I
don't know the comparative proportion) give definitions such as you
mentioned.

Topology by Munkres: "A function f is a rule of assignment r, together
with a set B that contains the image set of r. ..."

Real Analysis by Royden: "By a function f from (on on) a set X to (or
into) a set Y we mean a rule ..."

Algebra by Mac Lane and Birkhoff: "A function f on a set S to a set T
assigns ..."

Naive Set Theory by Halmos: "If X and Y are sets, a function from (or
on) X to (or into) Y is a relation f such that ..."

A Mathematical Introduction to Logic by Enderton: "A function is a
relation F with the property that for each x in dom F there is only one
y such that <x,y> in F."

Topics in Algebra by Herstein: "If S and T are nonempty sets, then a
mapping from S to T is a subset, M, of S X T such that ..."

Real Analysis and Probability by Dudley: "Informally, given sets D and
E, a function f on D is defined by assigning to each x in D one (and
only one!) member f(x) of E. Formally, a function is defined as a set f
of ordered pairs <x,y> such that for any x, y, and z, if <x,y> in f and
<x,z> in f, then y = z."

Dudley is a bit on the fence, but I'll give him to you. Perhaps not a
large sample, but 5/7 of the books require a function to have a
specified codomain.

--
David Marcus
.

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