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?
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.