A role for a non-symmetric equality relation in teaching mathematics? [closed]

Question

First, I will simply observe that it seems to be standard practice, in elementary set theory, to define relations to be sets of ordered pairs. If we had the option of introducing a "symmetric relation" to be not a particular kind of relation, but instead a set of unordered pairs, then the commitment to using a set of unordered pairs to represent equality would remove the option of having a non-symmetric equality.

Now, the whole point of this thread is that the option of having a non-symmetric equality might be a convenient thing to have, so the initial structural decision to represent all relations via sets of ordered pairs may have unanticipated benefits.

Now, let us suppose that students are provided with the following premise:

$ \forall t \in Q \left( t \neq 0 \Rightarrow \left( \frac{t}{t} = 1 \right) \right)$

Let us suppose that students are required to accept any conclusions that can be deduced -- via valid reasoning -- from the above premise.

Then, in particular, we can replace the conditional ...

$t \neq 0 \Rightarrow \left( \frac{t}{t} = 1 \right)$

... with its contrapositive, because the original conditional is logically equivalent to its contrapositive:

$\left( \frac{t}{t} \neq 1 \right) \Rightarrow t = 0$.

Therefore, the original premise that was provided to students is logically equivalent to the following:

$ \forall t \in Q \left( \left( \frac{t}{t} \neq 1 \right) \Rightarrow t = 0 \right)$.

Now, given that zero is a rational number, we have ... $ 0 \in Q$

So -- in asserting the original premise and obligating students to accept all conclusions that can be deduced from it -- teachers are requiring students to accept (via the substitution $t = 0$) the following statement as true:

$ \frac{0}{0} \neq 1 \Rightarrow 0 = 0$.

Now, one way to avoid the above difficulties is to revise the premise. In particular, if we have a set S, such that ...

$ \forall x \left( x \in S \iff \left( x \in Q \land x \neq 0 \right) \right)$

... then the teacher may formulate the following proposed alternative premise:

$ \forall t \in S \left( \frac{t}{t} = 1\right)$

We see a difference between two approaches:

1. Assert a potentially problematic statement as a premise to be accepted, and if any difficulties are encountered, then build a layer of interpretations and prohibitions to attempt to prevent the problems from being exposed, or to prevent people from giving serious consideration to the problems when they are exposed; and
2. Revise the premise to refrain from asserting what we don't wish to assert

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Now, all of the above was merely a prelude or introduction to the main point of this thread.

We could imagine having a non-symmetric equality relation, where we interpret the assertion ... $$t = u$$ ...standing alone as a premise to be accepted -- after we have replaced the letter $t$ with some term and replaced the letter $u$ with some term -- to mean that we are authorized, if we see the term $t$ in a statement, to replace it with $u$, but where the reverse substitution is not necessarily authorized.

Given such an approach to equality, we could imagine having:

$ \Bigl( \Bigl( (0, t) \in = \Bigr) \land \Bigl( (t, 0) \notin = \Bigr) \Bigr)$

... where we are making no distinction between the equality relation and its attempted representation as a set of ordered pairs, going so far as to replace the conventional sequence of symbols "$0 = t$" (analogous to the sequence [subject verb object] in a natural, spoken language) with the sequence of symbols: $(0,t) \in =$.

Now, suppose that we refrain from asserting ...$$\left( \frac{t}{t} = 1\right)$$

... unless we have $\Bigl( (t, 0) \notin = \Bigr)$.

Given a non-symmetric equality, we would be permitted to nevertheless have $$\Bigl( (0, t) \in = \Bigr)$$ ... and although it may be somewhat confusing given established habits, we could write that as: $$ 0 = t $$ ...even though we would be committed to remembering that the only reason we were authorized to assert ... $$\left( \frac{t}{t} = 1\right)$$ ... was that we have $t$ restricted to values such that $ t \neq 0$.

Question: Could a non-symmetric equality relation play a positive role in teaching mathematics, such as -- if nothing else -- by displaying in explicit form what some students might be silently thinking, and that they should understand the pitfalls of?

Answer 1

I believe the answer to this question is "No". Getting students to utilize the equal sign correctly is enough of a headache as it is; they do not understand regular equality.

This sounds like a potentially interesting project in mathematical logic, but I do not see any pedagogical uses.

Of course, I cannot support this answer with any evidence because you have just invented this concept now, and there have never been any experiments on using this to teach students. So I can only give you my "gut reaction" from my experience as a teacher.