Let X be any set(it could be finite or infinite),and let T={0,1}. Prove that |𝓟(X)| = |F(X,T)| where 𝓟(X) is the power set of X and F(X , T) is the set of all functions from X to T.
So, I have tried to formulate a bijective function between the two sets, in order to prove that they are in one-to-one correspondence, and that means that they have the same cardinality.
Intuitively, this makes sense, because I lets say we have a finite set: X= {1,2,3,4}. So the 𝓟(X)={{1},{2},{3},{4},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,2,3,4},{2,3},{2,4},{3,4},{2,3,4}...}
Essentially, there are 2^n elements in the power set. So there have to be the same amount of functions that map from the set X to T.
Now, for this we define a function that takes all the even numbers and maps them to zero; odd numbers that maps them to one. So every subset of X gives a function that provides this mapping. So every subset of X has a mapping to T.
How do I go about constructing and proving a bijective function between the set X and the set of functions from X to T? My main problem is constructing that function!