∅∅={∅}

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Suppose X and Y are sets. What is Y^X?
Y^X is the set of all functions from X to Y.

What is a function from X to Y?
According to the modern definition, f is a function from X to Y is simply a subset of X×Y satisfying

• Every element x∈X associates with a unique element f(x) in Y.

Put it in another way, a subset f of X×Y is a function if

1. No element x∈X associates with nothing in Y, i.e. f(x) always exists and
2. No element x∈X associates with two or more elements in Y, i.e. there cannot be two f(x).

Examples:

1. f(x)=a square root of x is a function from [0,∞) to [0, ∞).
2. f(x)=a square root of x is not a function from (-∞,∞) to [0, ∞) because there is no f(-1).
3. f(x)=a square root of x is not a function from [0,∞) to (-∞, ∞) because there are two f(1), i.e. 1 and -1.

If M is a set with m elements and N is a set with n elements (m,n≠0), then what is the size of M^N?
Easy. M^N has exactly mⁿ.
For instance, let M={0,1,2} and N={0,1}, then M^N contains the following 3²=9 elements:
f(0)=f(1)=0;
f(0)=0, f(1)=1;
f(0)=0, f(1)=2;
f(0)=1, f(1)=0;
f(0)=f(1)=1;
f(0)=1, f(1)=2;
f(0)=2, f(1)=0;
f(0)=2, f(1)=1;
f(0)=f(1)=2.

What is ∅^X if X≠∅?
∅^X=∅.
Take any element x of X, we see that it can associate with nothing! Therefore no function exists. Note that it matches with our common notation that 0ⁿ=0 if n≠0.

What is Y^∅ if Y≠∅?
Y^∅={∅}, the set which contains the empty set as its unique element!!!
∅=∅×Y is itself a function from ∅ to Y, why? Because there is no element in an empty set, hence (1) no elements in an empty set will corresponds to no elements and (2) no elements in an empty set will corresponds to two or more elements. Weird!! Counting the number of elements, we see that it matches our common notation that m⁰=1 if m≠0.

What is ∅^∅?
Using exactly the same argument as before, we know that ∅^∅={∅}.
Counting the number of elements, we have 0⁰=1! This equality is true only if we consider 0 as a counting number.