Is there a counter example to the reflexive property?

[barson90]

The reflexive property of equality just says that a = a: anything is congruent to itself: the equals sign is like a mirror, and the image it "reflects" is the same as the original.

- from "Ask Dr. Math" -- http://mathforum.org/dr.math/faq/faq.pr ... ssary.html

Are there any counter examples to this?

I was thinking of the Heisenberg Uncertainty principle, which states that one cannot "know" simultaneously both the position and velocity of anything very small, such as an electron.

Since average velocity = distance / elapsed time , and since distance requires knowledge of position, can we say that velocity does not equal velocity? I feel like there's a way to tie this all together to generate a v does not equal v situation, but I'm not really sure whether this argument would be logical and / or accepted in mathematics. (Perhaps instantaneous velocity has something to do with it -- the limit as h approaches 0 of [f(x + h) - f(x)]/h

I'd appreciate any thoughts on this, or any other ways of thinking about the reflexive property of equality.

 

[daon]

Just a few things I found wrong with the post:

1. If something doesn't satisfy reflexivity, then under current convention, the relation is not an equivlance one. For example "less-than" is not reflexive.
2. Distance does not require knowledge of position. A car keeps track of the distance it has gone, but doesn't know where it is. Unless you mean being the observer.
3. Not knowing the value of something could never imply inequality.

And a tip:

Don't try to disprove a definition. A definition is what gives something meaning, it's not something to find a counter-example for.

 

[DrMike]

Equality is reflexive. By definition, a = a for any a. By definition, if a ~ a for any a, then '~' is reflexive.

There are other relations that are relexive, eg "is the same shape as", or "likes the same movies as".

There are also plenty of relations that are not reflexive, and even many that are antireflexive (a ~ a for no a) ... inequality springs to mind, but there are others.