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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.

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    $\begingroup$ "Another student at my university back at home noted that $\implies$ denotes a biconditional in discrete math" -- I highly, highly doubt this. (Edit: Not that the student told you, but that it is true.) I would be shocked if anyone could show me a textbook with this (terrible) convention. $\endgroup$
    – Mees de Vries
    Jun 11, 2018 at 8:21
  • $\begingroup$ @MeesdeVries Hmm. So I can say that it does not denote, the biconditional. By the way, I saw your edit! $\endgroup$
    – Prime
    Jun 11, 2018 at 8:22
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    $\begingroup$ in my experience, $\rightarrow$ is used by students who have not yet taken a class on formal logic/proof writing, and doesn't have a precise mathematical meaning. $\endgroup$
    – Albert
    Jun 11, 2018 at 8:23
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    $\begingroup$ See Material conditional for references and discussion: the symbol has a long history and it symbolize a difficult concept with a never-ending philosophical discussion; this (in part) explains the difficulty to agree on a "standard". $\endgroup$
    – Mauro ALLEGRANZA
    Jun 11, 2018 at 8:36
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    $\begingroup$ You can find tens of similar questions (with good answers) on this site. The issue is that we have several concepts in place : the conditional (the propositional conncetive) : $\to$ or $\supset$; the derivability relation in a formal system : $\vdash$; the logical consequence relation : $\vDash$. An then we have the $\Rightarrow$ symbol: someone uses it in place of $\to$, someone uses it in place of $\vDash$. Conclusion: up to now there is no International Mathematical Symbolism Standardization Office... $\endgroup$
    – Mauro ALLEGRANZA
    Jun 11, 2018 at 9:23

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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).

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  • $\begingroup$ At my university, the PhD's are very set on not using them interchangeably. If I wrote $\rightarrow$ on an exam at my school, it would be points off, or a note to not use such notation in favor of $\Rightarrow$. However, at UND, the professor perhaps simply does not want student using $\Rightarrow$ without understanding how to properly use it. Since discrete math is open to computer science students and others, is that perhaps why you think they might only use $\rightarrow$ instead of $\Rightarrow$ ? Is it that students in discrete math are too mathematically immature? $\endgroup$
    – Prime
    Jun 11, 2018 at 8:36
  • $\begingroup$ No, I do not think that is the reason. What would it mean that they are not mature enough to use $\implies$ "properly" if they are instead using $\to$ to express the same thing? Notation has no intrinsic meaning. The only reason I'd see teachers taking points off for using the wrong arrow is that, once a convention is established, the ability to follow the convention demonstrates that you understand the convention and thus the material. (Consider a student who confuses $x = -1 \implies x^2 = 1$ with $(1/n) \to 0$.) That, or the teachers are just passing on a convention they were taught. $\endgroup$
    – Mees de Vries
    Jun 11, 2018 at 8:46
  • $\begingroup$ In an above comment, @Mauro ALLEGRANZA seems to differentiate between the two notations as there being an actual difference. Apparently $\rightarrow$ is more for the logician and $\Rightarrow$ is more mathematically motivated and reserved moreso for theorems and lemmas. $\endgroup$
    – Prime
    Jun 11, 2018 at 8:50
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    $\begingroup$ @SincerelyPrime, I would argue that this differentiation is similar to the one I make in the second paragraph of my answer. $\endgroup$
    – Mees de Vries
    Jun 11, 2018 at 9:00
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    $\begingroup$ while books and papers in logic made heavy use of a symbol for implication, I suspect the current distinction between what mathematicians use and what logicians use originates from inertia based on earlier typographical issues. $\endgroup$
    – Dave L. Renfro
    Jun 11, 2018 at 9:04

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