Is using the if then symbol okay to use in regular proofs, and at what point is using symbols too much only to be reserved for set-theory?

Question

In calculus, we are proving statements since the course is a proof-based course. One such proof is the proof that $|ab|=|a||b|$. The proof makes sense but seems like it ought to be seen in set-theory, not calculus. The final statement is $\therefore\forall$ $ a,b\in\mathbb{R}$ $|ab|=|a||b|$. While this makes sense, is this okay for a calculus proof? Also, on another proof about absolute value, I would like to write $\therefore a\le k \rightarrow|a|\le k$. This would mean "therefore if/since $a\le k$, then $|a|\le k$". Is this a "proper" use of $\rightarrow$? In other words, does this fit within the scale of calculus or should it be kept to set-theory? Thanks in advance for any help you might be able to offer.

Answer 1

Upgrading my comment to an answer.

As a matter of principle I recommend trying to avoid using symbols from mathematical logic in prose outside of the symbolic definition of a mathematical object. This is because it can make your proofs overly terse.

In other words, write out "for all" in the sentence "For all $x,y\in\mathbb R$, we have $x^2+y^2\geq 0$.", but you can still write $$S=\{(x,y)\in\mathbb R^2 \mid x^2+y^2\leq t^2\text{, } \forall t\in T\}\text{.}$$

However, I will add that it is not logically incorrect to use symbols in a proof, as long as the meaning is understood by the reader. That is a key consideration, as all writing, even proof-writing, is an exercise in communication.

To your next point, on the proper use of "$\rightarrow$" in the statement "$\therefore a\le k \rightarrow|a|\le k$", this symbol is often used for convergence, so in this context, if you really must use a symbol for implication, better to use a double arrow, e.g. $\implies$.

[Edit: note that as for the actual mathematics, I'm sure you're aware that in a vacuum, $a\le k \implies |a|\le k$ is a false statement, as in the case $a=-5, k=2$, for example.]