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 $$\Rightarrow$$ instead of "if" and "then":

"It rains $\Rightarrow$ The sky is gray."

In the end, we will have to write a whole lot of stuff, so using the symbol $$\Rightarrow$$ is totally useful.

Constructing new knowledge
Let's say we have two implications, say

"It rains $\Rightarrow$ The sky is gray."

and "The sky is gray $\Rightarrow$ It is darker than usual."

Then we can deduce:

"It rains $\Rightarrow$ 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?