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.
Best Answer
There's two ways to use $\Rightarrow$:
As notation for the relevant function $\{0,1\}^2 \rightarrow \{0,1\}$.
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.