Ambiguity or Flexibility?

Date: 10/22/2003 at 20:43:43
From: Steve
Subject: The not equal sign

While contemplating the use of the not-equal sign (=/=), I 
observed a certain ambiguity. 

On one hand, if we say that for some x, x =/= 3 then what we are
actually saying is that x can never be 3.

On the other hand if we write that for some a and b, a^b =/= b^a, 
what we mean is that there exists at least one case where these two 
expressions are not equal.  That is, we say that they are not always 
equal to each other (in fact, most of the time) but we don't claim 
that they are always different, and sure enough 2^4 = 4^2.

So it seems that sometimes we use the not-equal sign to state an 
absolute inequality (two expressions always have a different value), 
and sometimes we just mean to say that two expressions are not 
always equal.

Can you please clarify this apparent ambiguity?  How do we know which 
meaning is being used?

Basically, the question is whether "=/=" means "never equal" or "not 
always equal" or both?

Date: 10/22/2003 at 22:51:18
From: Doctor Peterson
Subject: Re: The not equal sign

If I write

  1 =/= 2

it means 1 IS NOT equal to 2, which I suppose you could take to mean 
1 IS NEVER equal to 2, since the inequality is always true.

But if I write

  x =/= 1

it just means x is not equal to 1.  It doesn't mean x is never equal 
to 1, because obviously it can be, if you let x equal 1.  It doesn't 
mean x is not always equal to 1, because that says nothing.  It just 
means that for whatever x I have chosen, or whatever x I am allowing 
you to choose, it is not equal to 1.  Nothing more, nothing less.

So the sense in which something is not equal depends on context (what 
can vary, how the inequality is being used).

The ambiguity you note (which has nothing to do with the symbol, 
but only with the context in which it is being used) is true of 
equations.  There are several ways in which an equation can be used.  
It can be an identity (always true) like

  ab = ba for all a and b

or it can be an equation to be solved (to be made true by choosing the
right value, which may or may not be possible) like

  x^2 + 2x + 1 = 0

or it can serve as a definition (always true because it's telling us
how to interpret something) like

  f(x) = x + 1

and probably it can be used in other ways that don't occur to me.

Your example of a^b =/= b^a does not say "they are not always equal", 
unless you say, in context,

  a^b =/= b^a  for some a and b

It is the whole statement, not just the inequality itself, that says 
"not always".  The inequality just says "for a particular a and b, 
they are not equal".  It does not assert that such a and b exist, 
unless you say so.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/23/2003 at 04:52:30
From: Steve
Subject: Thank you (The not equal sign)

Thanks for your quick reply.

I guess I was struggling in vain to find that "ultimate" one meaning,
but your example clearly shows that we are used to living with the
same ambiguity with the equal sign.

Date: 10/23/2003 at 08:34:58
From: Doctor Peterson
Subject: Re: Thank you (The not equal sign)

Hi, Steve.

Actually, this sort of "ambiguity" is inherent in any communication, 
in English as well as in math symbols.  The words we use have slightly 
different meanings in different contexts; it is the way we put words 
together that determines the meaning of a sentence.  I would prefer to 
call this "flexibility" rather than "ambiguity": we can use a symbol 
like "equal" or "not equal" for different purposes, which makes it 
more useful.  The meaning is always clear--well, at least as long as 
we write clearly!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/