Why does the symbol for implication vary from class to class, university to university?

Question

Some (neccesary) background and motivation for my question:

In some of my undergraduate math classes, specifically my proofs course, I have been required to use the $\Rightarrow$ symbol to denote any implication. In fact, my professor told me that we reserve $\rightarrow$ for "different things in math, like limits". This was at the university that I currently study at and many of my math professors use this notation. This is what I am used to to denote an implication of any sort. This class is specific to the mathematics major.

This summer, I decided to take a discrete math class online with the University of North Dakota. In submitting an assignment very similar in scope and content, even identical to the assignments at my home university, the professor told us that we are to use $\rightarrow$ as $\Rightarrow$ is reserved for much more formal mathematics.

This assignment was identical in content and scope as an assignment I had in my proofs class

Another student at my university back at home noted that $\Rightarrow$ denotes a biconditional in discrete math, but I had been taught since high school and all the way up through university mathematics (I am a junior math major now) that the biconditional is denoted with $\iff$.

Even if $\Rightarrow$ did denote a biconditional in discrete math, then why does $\leftrightarrow$ exist?

My question:

Why do these differences exist, what are these differences, and why do they seem to be particular to discrete mathematics?

I would appreciate any and all insight.

Answer 1

Why do these differences exist, what are these differences, and why do they seem to be particular to discrete mathematics?

Of course I can't speak to your specific context, but I very strongly suspect that it is simply that notation is used slightly differently in different places, each with their own convention. To students convention is often presented as fact (and sometimes this is not even bad -- e.g., the convention that multiplication binds more strongly than addition is so strong that it might as well be a fact).

The only time I have seen a strong distinction between $\implies$ and $\to$ is in logic, where $\to$ is the syntactical implication, used in formal sentences like $\forall x(P(x) \to Q(x))$, while $\implies$ is used for meta-language implication, the language in which you state theorems and proofs. E.g:

Theorem: For $\Gamma$ a set of formulas, and $\phi, \psi$ formulas, we have that $$ \Gamma \vdash \phi \to \psi \implies \Gamma,\phi \vdash \psi. $$

Outside of this I have only ever seen $\to$ and $\implies$ used more or less interchangeably (although typically consistently within a single text).