[Phys-L] Re: arithmetic

["John S. Denker" <jsd@AV8N.COM>]

On 02/24/05 09:21, RAUBER, JOEL wrote:
> Reflexive  A=B  => B=A is the symmetric property isn't it?  Or are you
> referring to something else?

     Reflexive:  a = a
     Symmetric:  if a = b, then b = a
     Transitive: if a = b and b = c, then a = c

These three properties are mutually independent, i.e.
none is a consequence of the others.

Together, they *define* what we mean by equivalence
relation.

http://mathworld.wolfram.com/EquivalenceRelation.html

Remembering the _names_ of the properties is a pain.
The symmetric property looks symmetric, but the
reflexive property looks symmetric also, so that's
not much help.  And a=b looks like a "reflection"
of b=a, so that's not much help, either.  Here's
a mnemonic that's helpful if you speak French or
Spanish:  A _reflexive_ verb uses some version
of "se".  For example
   French:          Ils se sont réveillés.
   Literal English: They woke 'themselves' up.
   (Proper English: They woke up.)

   French:          Il se lave les mains.
   Literal English: He washes 'his own' hands.

So the mnemonic is:
   reflexive --> se --> 'self' --> a equals itself
   i.e. a=a.

=============================

You can define lots of different equivalence relations.
For instance, in a deck of cards, you can have
  -- is-same-suit
  -- is-same-rank
  -- etc.

So for example, the ace of hearts is equivalent to
the ace of clubs w.r.t rank.  The 9 of clubs is
equivalent to the 3 of clubs w.r.t suit.


Sticking to numbers, the numeral 8 is not equivalent
to the numeral VIII, if we just look at the _numerals_
... but they are equivalent as to the _number_ they
represent.

I mention this because a lot of students seem to have
picked up the mistaken idea that "equals" somehow
ought to mean "absolutely identical in all respects"
... which is almost never what it means.

Also defining equality in terms of reflexive, symmetric,
and transitive properties fits in nicely with what I
said yesterday about defining multiplication, division,
and even addition in terms of how they behave w.r.t thge
identity, inverse, distributive property, et cetera.