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Suppose

*X*and

*Y*are sets. What is

*Y*?

^{X}*Y*is the set of all functions from

^{X}*X*to

*Y*. From the last post, we see that if

*X*has

*n*elements and

*Y*has

*m*elements, then

*Y*has

^{X}*m*elements.

^{n}What is 2

*?*

^{X}If we write like that, we identify 2 with a set with two elements, in particular we can consider 2={0,1}. Therefore 2

*is simply a set of functions that for each*

^{X}*x∈X*,

*f(x)*takes up values as 0 or 1.

How do we describe a function in 2

*?*

^{X}Since

*f(x)*can only have two values, we can describe the function

*f*using the preimage of 0, i.e.

*f*={

^{-1}(0)*x∈X*:

*f(x)*=0}. In other words, each function in 2

*corresponds to exactly one subset of*

^{X}*X*.

2

*is sometimes used to denote the power set of*

^{X}*X*, right?

A power set of

*X*, usually denoted by P(

*X*), is the set of all subsets of

*X*. Because there exists a correspondence between the functions in 2

*and the subsets in P(*

^{X}*X*), so we sometimes really consider them identical.

For example, if

*X*={a,b,c}, then P(

*X*)={∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}} and the eight functions in 2

*are*

^{X}f(a)=f(b)=f(c)=1;

f(a)=0, f(b)=f(c)=1;

f(b)=0, f(a)=f(c)=1;

f(c)=0, f(a)=f(b)=1;

f(a)=f(b)=0, f(c)=1;

f(a)=f(c)=0, f(b)=1;

f(b)=f(c)=0, f(a)=1;

f(a)=f(b)=f(c)=0.