Some thoughts about Yes and No
Everybody knows Yes and No. Here are how it works:
A No with Yes equals No; A Yes with No equals No.
A No with No equals Yes; A Yes with Yes is still Yes.
Furthermore, we can arrange any number of Yes and No in a sentence though we don’t do this in reality.
What do you see?
To simplify the notation, I will use ‘Y’ to represent ‘Yes’ and ‘N’ to represent ‘No’. Also, let’s say \(a\) and ‘Y’ are the same and so do \(b\) and ‘N’. Asterisk \(*\) stands for how Yes and No are arranged. Now we can rewrite the mechanism of Yes and No as:
\[a*b=b*a=b\] \[a*a=b*b=a\]The first equation exactly stands for the commutative law! Suppose $S = {a,b}$, then \(a*b=b\) and \(a*a=a\) mean that $a$ is an identity element of \(S\). Then property \(b*b=a=identity\) plus \(a*a=a=identity\) means that every element of set \(S\) has a unique inverse. If you have basic idea of group theory, you should notice that \(S\) is exactly an abelian group(To see this, we only need to verify the associative law which can be easily done). It is remarkable that this simple pair of words actually forms an abelian group which plays a irreplaceable role in modern algebra.
In addition, ‘positive’ and ‘negative’(-1 and 1) can also be regarded in this way by multiplication, i.e. \((-1)\times 1=1 \times (-1)=-1\) and \((-1)\times (-1)=1\times 1=1\). Apart from multiplication, addition of \(\{0,1\}\) are able to behave in the same way.