#isomorphism

Sinceramente, una de las fotografías a las que más afecto le tengo.

First, going from Korean to Japanese, you will notice that the function (translation) is bijective (word-for-word) because there exists a map between the two sets (sentences) where each element (word or word-part) in the first set (Korean) is paired with exactly one unique element of the second set (Japanese), each element of the second set is paired with exactly one unique element of the first set, and there are no repeats. Moreover, the mapping preserves the relations between elements in the sets and is, therefore, a homomorphism. In other words, the translation preserves the grammatical part of speech of each word or word-part (i.e., the direct object remains the direct object). Thus, the translation of this sentence from Korean to Japanese is a grammatical isomorphism. Obviously, the same can be said of the Japanese-to-Korean translation.

On the other hand, the translation from Japanese to English is not even a proper function because there exist no elements (word or word-parts) in the codomain (English) for two elements in the domain (Japanese): the topic particle は and the direct object particle を. Likewise, in the translation from English to Japanese, there exist no elements in the codomain for two elements in the domain: the definite article "the" and the possessive adjective "my." Additionally, there are two elements in the domain that map onto two elements each in the codomain: "went" to 行き and ました, and "with" to と and 一緒に. There is no function, let alone isomorphism, between English and Japanese. By extension, there is none between English and Korean, either.

나는 친구하고 같이 공원에 점심을 먹으러 갔었어요.

私は友達と一緒に公園に昼ごはんを食べに行きました。

As opposed to English:

the pigeonhole principle, as applied to the enter() and data() verbs of d3.js