# 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 ${\displaystyle \Rightarrow }$ instead of "if" and "then":

It rains ${\displaystyle \Rightarrow }$ The sky is gray.

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

## Constructing new knowledge

Let's say we have two implications, say

It rains ${\displaystyle \Rightarrow }$ The sky is gray.

and

The sky is gray ${\displaystyle \Rightarrow }$ It is darker than usual.

Then we can deduce:

It rains ${\displaystyle \Rightarrow }$ 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?