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(Growing lazy, not a good sign....)

I have seen a proof, saying that 1=0!?

Which one do you mean '"1=0!"?' or '"1=0"!?', uh?

You got the trick. Why do we use "!" for factorial?

I am not a historian, I don't know why Christian Kramp choose

*n*! to represent factorial.

Why does 0!=1?

Lets first consider the definition of factorial. The n! is the number of ways to permute n objects. In the set theory language,

*n*! is the size of the set of all bijective maps on a set of

*n*elements.

Let

*A*be a set. A bijective map or a bijection or a permutation is a subset F of

*A*×

*A*such that

- For any elements a∈
*A*, there exists a unique F(a)∈*A*such that (a,F(a))∈F. - For any elements a∈
*A*, there exists a unique F^{-1}(a)∈*A*such that (a,F^{-1}(a)∈F.

On {x,y,z}, we have six permutations, right?

Yes. They are

- (F(x), F(y), F(z))=(x, y, z)
- (F(x), F(y), F(z))=(x, z, y)
- (F(x), F(y), F(z))=(y, x, z)
- (F(x), F(y), F(z))=(y, z, x)
- (F(x), F(y), F(z))=(z, x, y)
- (F(x), F(y), F(z))=(z, y, x)

On empty set ∅, ...?

Note that ∅ equals to and is the only subset of ∅×∅. There is no way we can violate the two conditions, and so ∅ is the bijection on ∅. Therefore 0!=1.