intoononto

Intoononto is a neologism that appears in some mathematics education discussions and online forums to discuss the relationship between a function and its codomain. The word is a portmanteau of into and onto, two standard descriptors of how a function maps domain elements to codomain elements. Because into describes that the image of the function sits inside the codomain and onto says the image covers the codomain, intoononto is used informally to signal how these two aspects relate in a given mapping.

Usage varies: some writers use intoononto as a mnemonic for a bijection, viewing the term as an emphasis that the function is simultaneously into (its image lies in the codomain) and onto (the image equals the codomain). Others deploy it to remind students that onto does not necessarily imply into in contexts where the codomain differs in structure, or to discuss functions where one property holds but not the other. In other contexts, intoononto is simply a playful label with no formal standing in standard textbooks, and its meaning must be inferred from context.

Examples commonly cited include the identity function on any set and any permutation of a set, which are bijections and hence satisfy both into and onto. By contrast, non-bijective functions illustrate cases where only one property holds, depending on how the codomain is chosen.

Origin and reception: The term first appears in informal online discussions in the 2010s and has not been adopted into formal curricula; it is mainly used as a mnemonic aid or curiosity.

See also injective, surjective, bijective, function, mapping.