Be smart - learn mathematics!/Implications and equivalences

Now I hope you aren't tong-tied after "implication"; what it is, is an "IF, THEN" relationship. A simple example: If it rains, then the sky is gray. This is already an implication. In this case, "it rains" implies "the sky it gray". Have you got that? Let's do another one: If you sleep, then your eyes are closed. Now you may think that's boring, but what we can do with it is to construct new knowledge - just out of the blue! This is what scientists do, and it is just exciting. Note that in order to be able to write quicker, we write instead of "if" and "then": It rains The sky is gray. In the end, we will have to write a whole lot of stuff, so using the symbol is totally useful. Constructing new knowledge Let's say we have two implications, say It rains The sky is gray. and The sky is gray It is darker than usual. Then we can deduce: It rains It is darker than usual. Why? Because whenever it rains, the sky is gray, and therefore it is darker than usual. It's true, isn't it?