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.
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.