View source for 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:

{{quote|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:

{{quote|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 <math>\Rightarrow</math> instead of "if" and "then":

{{quote|It rains <math>\Rightarrow</math> The sky is gray.}}

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

== Constructing new knowledge ==

Let's say we have two implications, say

{{quote|It rains <math>\Rightarrow</math> The sky is gray.}}
and
{{quote|The sky is gray <math>\Rightarrow</math> It is darker than usual.}}

Then we can deduce:

{{quote|It rains <math>\Rightarrow</math> 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?

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