Re: 2 pi i = 0

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Michael's comment:

> If we have y = sqrt(x^2+25) then y has a relation to x, but y is not a
> function of x.  The graph of y versus x is a circle centered on the
> origin with radius = 5.  It does not pass the 1-to-1 test, or stated
> another way the graph does not pass the "vertical line test."

I hope the definition of a circle has not been redefined, too.  The
equation Michael wrote above describes what I would call a hyperbola,
centered on the origin and opening vertically.

> In my school days y = sqrt(x^2 + 25) was every bit as much a function as
> y = 2x + 5.  The former was a double-valued function except single
> valued at x = +or- 5.  The latter was a single-valued function.  We did
> use the "vertical line test" but only to determine if the function was
> multi-valued or not.
>
> Bottom line, it appears to me the old-timer definition of function is
> the same as the current definition of a relation.

Apparently, I'm a mid-timer since the definition I was taught for a
function is neither of the ones discussed so far by Michael (although
it has elements of both).  I was taught that a function was
(synonymous with the terms 'correspondence' and 'mapping') any rule
that associated a *unique* value for each argument.  The mapping
could certainly be many-to-one, but it could *not* be one-to-many
and still be said to be a function.

Also, I was taught that a relation was something that expressed a
relationship between things.  Relations are usually labeled by a
symbol between the things said to be related.  For instance the
symbols < , = , > , = (mod 7) , etc. denote the relations 'less
than', 'equals', 'greater than', 'equals mod 7', etc.  If R is a
relation then A R B means that A is related to B via the
relationship denoted by R.  R could also express some other
usually nonmathematical relationship as well as explicitly
relationships that have conventional mathematical applications.
For instance, R could signify 'is the son of', 'is the sibling of',
'is the ancestor of', 'is the same color as', 'is not similar to',
'is contained in', 'is the antiparticle of', 'chemically reacts
with', etc.

Whenever a relation R has the property that A R B necessarily implies
that B R A for all A & B then R is said to be a *symmetric* relation.
Whenever a relation R has the property that A R A for all A then R is
said to be *reflexive* relation.  Whenever R has the property that
it is necessarily the case that A R B and A R C implies that A R C,
then relation R is said to be a *transitive* relation.  If a relation
is simultaneously symmetric, reflexive, and transitive then it is
said to be an *equivalence* relation.  If A & B are related by an
equivalence relation then they are said to be equivalent (under that
equivalence relation).

David Bowman

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.