Empty set vs empty string vs nothing and Zen

From my explanations of the set theory to my daughter yesterday.

A set is a container, an empty string (ε or nothing) is an element. A set with nothing inside is not empty, because there is nothing inside. A set is empty (∅) if it does not have anything inside.

In python:

$ python
>>> nothing=''
>>> set=[nothing]
>>> set
>>> set.append('')
>>> set
['', '']
>>> set.remove('')
>>> set
>>> set.remove(nothing)
>>> set
>>> emptyset=[]
>>> emptyset

Doing nothing or not doing anything are two different things.

Why does x^0 equal 1?

Since basic school we know, that [math]x^0=1[/math], but only recently when our kids began to study mathematics we asked ourselves “why?”.

I came up with this explanation:

[math]x^0=x^{n-n}=x^n \times x^{-n}=\frac {x^n} {x^n}=1[/math]

Also a good test for the Easy LaTeX WordPress plug-in.

And here’s jsxgraph plug-in test:
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