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A lecturer mentioned that a common mistake people make in assignments is the incorrect use of the implication notation, $\Rightarrow $. I would like to clarify the correct use of the symbol as I am responsible for marking some first year assignments this term, and have been advised to deduct marks if students make this 'mistake'.

The symbol should be used, I am told, only when making a logical statement $A\Rightarrow B $, i.e. when the truth value is unknown. In other situations where we know $A $ is true, we should use the therefore symbol $\therefore $. So, for example, a mark would need to be deducted for the following answer:

Q: If $(a_n),(b_n) $ are positive, bounded real sequences, then $(a_nb_n) $ is also bounded.

A: $(a_n),(b_n) $ bounded $\Rightarrow $ $a_n <A$ for some $A$ for all $n $, $b_n <B$ for some $B $ for all $n $ $\Rightarrow $ $a_nb_n <AB $ for all $n $ $\Rightarrow $ $(a_nb_n)$ is bounded.

A mark would be deducted since $(a_n),(b_n) $ bounded was a hypothesis of the question. However, I see this as pedantic, since if I add the following line to the proof then it will be correct:

And since $(a_n),(b_n) $ bounded is assumed, it follows that $(a_nb_n) $ is bounded.

Am I right to say that this makes the argument 100% correct? I will add that the line need not be added in the first place, because given the context (an assignment answer), it is clear that this is what the author intended.

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  • $\begingroup$ I'd recommend speaking to the lecturer so that you are on the right page. You are effectively an employee of the lecturer and clarification of a task from your boss is normal in any workplace. $\endgroup$ – Ian Miller Oct 7 '16 at 10:56
  • $\begingroup$ Are you sure you picked a good example? I'm as pedantic as they come and I see nothing wrong with a proof that looks like $A\implies B\implies C\implies D$. $\endgroup$ – Git Gud Oct 7 '16 at 11:12
  • $\begingroup$ There was a similar longer proof in the handbook, with the note 'The implication sign is their only attempt at connecting statements and it is used incorrectly: “P ⇒ Q” means “if P then Q”, but we know the predicates are true. So we should use “thus”, “hence”, “therefore” etc.' $\endgroup$ – Szmagpie Oct 7 '16 at 11:15
  • $\begingroup$ @Szmagpie Some people use $\implies$ instead of "then", "therefore", etc. This is wrong. This isn't being done in the example you've given though. $\endgroup$ – Git Gud Oct 7 '16 at 11:18
  • $\begingroup$ @GitGud I guess the motivation of my question is to compare the answers '$A\therefore B $' and '$A\implies B $, and $A$ is assumed, $\therefore B$'; then if they are equivalent, I argue that the latter half is redundant given the context, and as a result, $A\implies B $ is acceptable $\endgroup$ – Szmagpie Oct 7 '16 at 12:30
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I would not deduct any marks for the first answer.

What is an implication? It simply says "If A is true, then B is true". This is symbolically written as $A \implies B$.

When the implication is false, there is some object having property $B$ that does not have property $A$.

In the implication in question, it is clear that the author knows the context he is working in, and does not need another redundant statement to clarify to a well-read instructor that he is aware of the context. Therefore, there is nothing wrong with the logic of the question, I would detest a deduction of marks.

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There's two ways to use $\Rightarrow$:

  1. As notation for the relevant function $\{0,1\}^2 \rightarrow \{0,1\}$.

  2. As a syntactic ingredient in proofs.

Your lecturer is saying she doesn't like (2), which is fair enough. I wouldn't go as far as to call it "wrong"; that's too strong of a word in this context.

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