I was thinking about how we should interpreting a simple mathematical expression and the concept of congruence and equality. When working with algebra, we don't really make a distinction between the two and that is a huge source of confusion for people who are first learning mathematics.
Would you say that there is a logical distinction between equivalence and equality, and should we make it clear to learners? I'd define the two as:
- equality: If $A = B$, then the left is the same as the right
- equivalence: if $A\equiv B$ then expression A always produces the same output as expression B
Equivalence
Consider the expression:
$$ \left|{1\over x- 2}\right| $$
I'd say it's equivalent to:
$$ {1\over\sqrt{x^2-4x+4}} $$
Because no matter what I decide to choose as the input value for $x$, I'd always get the same output for each expression. So I'd write this as (this is a non-standard use, I'm just using this as another symbol):
$$ \left|{1\over x- 2}\right| \equiv {1\over\sqrt{x^2-4x+4}} $$
Naively, the use of equals here suggests that you can solve for $x$ in this expression.
Equality
If I replaced the equivalence above for an equality and asked somebody to solve for $x$, they'd quickly end up reducing the expression to $0=0$, because they are equivalent. If however, I told them to find $x$ in the following expression over the real numbers:
$$ \left|{1\over x- 2}\right| = {x\over 3} $$
We can actually make progress towards this by working with the left hand side and the right hand side separately, we can say:
$$\left|{1\over x- 2}\right| \equiv {1\over \sqrt{(x- 2)^2}} $$
Because these are equivalent, we can replace the left for the equivalent expression.
$$ {1\over \sqrt{(x- 2)^2}} = {x\over 3} $$
We can substitute an expression for any equivalent expression. In the case of a simultaneous equation for example, we can also substitute an equal expression too.
Equality as an Operator
I also consider equality to be similar to operators like $+, -$, etc. The = operator says "this statement is true" (similar to the programmers ==). So writing something like this becomes valid too (I've just put the brackets in to make this explicit):
$$ \left(\left|{1\over x- 2}\right| = {x\over 3}\right) \equiv \left({1\over \sqrt{(x- 2)^2}} = {x\over 3}\right) $$
Then it could be explained that instead of writing the equivalence sign, we alternatively just put things on a new line. This would force learners to use equivalence when they mean equivalence, and equality when they mean equality.
Summary
- Is there a good reason why we don't make a distinction between $=$ and $\equiv$, or does it just come down to laziness?
- In a first course on algebra, how would you reconcile the two different uses of the = sign that I've outlined above?
