Number of functions from one set to another

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SUMMARY

The number of functions from the set {1,...,n} to the set {0,1} is definitively 2n. Each element k in the set {1,...,n} can be mapped to either 0 or 1, resulting in two choices per element. Since there are n elements, the total number of functions is calculated as 2n. This conclusion can be formally proven using combinatorial principles.

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Homework Statement


How many functions are there from {1,...,n} to {0,1}? and prove.


The Attempt at a Solution


For any k in {1,...,n} we can map k to either 0 or 1. So for any k there are two possible images. Since there are n kinds of k (from 1 to n) then there are 2^n possible functions


I'm not sure how to formally prove this, or what kind of proof to use at least.
 
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I would call that a proof. How formal do you want to be?
 

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