Hash function - two different meanings?

My understanding of "hash function" is this: any function from a set A to a set {0, 1, 2, ..., n}, where n is a non-negative natural number. Nothing else is inherently part of what it means to be a "hash function". Both of your examples - and many other examples - consistute "hash functions", since they map things to a subset of non-negative integers.

The way in which a "hash function" is applied to a problem is also not part of the definition.

The term "hashing" generally covers both of the above meanings; as other answers point out, the operations are similar. Also, the two processes are generally used in tandem -- one is not really useful without the other.

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