7

I have recently come to discover Metamath.

Supposedly the language is one that a computer may proof-check.

I then began to look at concepts that I am familiar with, and decided to look up the pigeon hole principle.

In their proof, in line 5 I came across a use of inequality that I am uncomfortable with.

Supposedly given two sets $A$ and $B$, "$A\neq B$ iff $\lnot A = B$".

In particular, they are using this for the empty set: A is not equal the empty set if and only if not A is equal to the empty set.

I can find an easy counterexample to this statement.

Let $U = \{1,2,3,4,5,6\} , A = \{1,3,5\}, B = \{4,6\}$.

Clearly $A\neq\varnothing$, and $A \neq B$.

However, $ (\lnot A) \neq \varnothing$, and $ (\lnot A) \neq B $

Is there a particular case where the definition meta-math gives for the inequality definition valid? Other that A being equal to the universal set?

Update: I have added the use of some symbols, and corrected a few mistakes I have caught in the process of editing this post. Daniel has made an excellent input that the problem is probably in the syntax of the language and the ambiguous use of precedence.

StudentofEuler2718
  • 486
  • 6
    I think you misinterpret that. It's not "(Not A) is equal to the empty set" using the awkward terminology "Not A" for the complement, but "Not (A is equal to the empty set)". – Daniel Fischer Aug 06 '13 at 19:55
  • That would make sense, however why are they not using the parentheses?

    Ie, A not equal to B iff Not(A = B).

     – StudentofEuler2718 Aug 06 '13 at 20:01
  • 5
    Probably because (apparently) the precedence rules of the language are such that no parentheses are necessary there. Not that I find that a particularly fortunate design. – Daniel Fischer Aug 06 '13 at 20:05
  • 1
    "$\neg$" is an operation on propositions, not on sets. I'm not aware of "$\neg A$" being used to denote the complement of a set $A$. Unless that usage appears here, this isn't ambiguous since "$\neg A$" by itself has no meaning. – Noah Schweber Mar 21 '18 at 23:18

1 Answers1

4

In Metamath (specifically, in the syntax that was chosen for the set theory database set.mm, which is not built in) a few common infix operations are never surrounded by parentheses. That includes equals, is member of, and is subset of. So it is not ambiguous.

MarnixKlooster ReinstateMonica
  • 5,606
David A. Wheeler
  • 41
  • 1
    Note that Metamath works on strings of characters-- it does not even have the concept of parentheses built in. Given the choice to write A = B everywhere, or to everywhere write (A = B) even when the parentheses are superfluous for a human reader, the designer chose the first option. – MarnixKlooster ReinstateMonica Mar 12 '14 at 07:15