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I believe I understand what a function and relation are, but what is a mapping? At first I thought the term was synonyms with relation, but after looking it up, I’m thinking it could be more or less the same thing as a function.

Does a “mapping” or “map” imply a relationship with one output value per input?

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In most cases "map" or "mapping" is synonymous with "function".

You do need to be a little careful, because some authors will reserve the term "map" for functions with particular properties they are interested in. For instance, it's not uncommon in topology for "map" to mean "continuous function". People who are doing this should say so explicitly, so unless the author tells you otherwise, "map" just means "function".

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