Negation and notation
This one’s going to be pretty quick. Here’s a puzzle. Suppose you have functions $f$ and $g$, and you want to make it clear that $f(a) = g(b)$ and $f(-a) = g(-b)$ in a single terse line of notation. How do you do it? Note that we haven’t defined any other operations on $a$ and $b$ besides negation.
@expand@ Small hint
If you were to write $f(\pm a) = g(\pm b)$, that would actually say four things:
- $f(a) = g(b)$
- $f(a) = g(-b)$
- $f(-a) = g(b)$
- $f(-a) = g(-b)$
Of these four things, you only want to express #1 and #4.
@expand@ Bigger hint
If you were to write $f(\pm a) = g(\mp b)$, that would only give you #2 and #3 as written above. Almost, but not quite!
There may be better ways to do this, but here’s how I’d write it:\[f(\pm a) = g(\mp (-b))\]
Is it kind of ugly? Yes, but that’s the point. The notation is humorously inconsistent, and we can take advantage of that.
@/expand@tags: math - musings