I am following a Coursera course called "Introduction to Mathematical Thinking" by Keith Devlin. There is a lecture in the course by the name of Implication. In that video, while deriving the truth table for =>, few things were unclear to me.
- Let's just say a $\Rightarrow$ b. If speaking of genuine implication, studying mathematical thinking implies my math improves. So, we are assuming already that "Studying mathematical thinking" = True. And by definition, "My math improves" = True. However, I take the condition where "My math improves" = False, makes the whole statement false. So, we have (T,T)=T; (T,F)=F.
- Now, in the lecture, what they do is take a condition a $\nRightarrow$ b. In the next line, they state an obvious statement, where they tell that for this statement to be True, even though a = True, nevertheless b = False. The next line, however, is the one that concerns me. They conclude that apart from the case (T,F) = T for a $\nRightarrow$ b, all other cases are False. But my concern is that I understand this conclusion for (T,T) case. But how can you say that for (F,T) and (F,F) in the case of a $\nRightarrow$ b????? (In the next line they say that a $\nRightarrow$ b = ~(a $\Rightarrow$ b). I understand that bit.)
