Inverse image of a function
October 31, 2010 § Leave a comment
The first post of the blog is about sets and applications. The problem could be easily explained by saying that the inverse image of a function is actually the image of the “inverse” of this function, but let us say that things are not so, cause there are rules, in mathematics, which state that a function can have an inverse only if it respects some standard. Those standards are not met by every function (it should be bijective), but a function does not need to meet those standards to have an inverse image. There is a post in google docs which explains the problem in mathematical terms; the inverse image of a function results in a subset of the domain of the function, so the two exercises are correct by definition. However I do not think you can understand clearly the idea from a mathematical explanation so I am going to repeat the problem in a more specific manner.
There are many academics of the communication who have done a functional research about languages, and who have arrived to some specific conclusion, which I cannot explain in this post. However those assumptions let you define mathematical functions in communicative terms, as if they were operation on elements of a language. The language could be constituted by various elements, even not the spoken language; I will use two metaphors to explain the mathematical argument of this post, one with the language of design, and the second with the language of narration.
They are both different languages from the natural one, which we speak every day, even though both have a syntagmatic and paradigmatic side. The syntagmatic side of a particular design, for instance, associates to a certain content, its typography and graphical elements, as the violet image at the beginning of this post, which is made by a “comic” character setting and a subsequence of images from right to left, a Japanese order. Another kind of design could be the one of a bank report, which usually is more formal in the typography and uses text and images from left to right.
In the language of narrative, and at most in the narrative for tabloids, a syntagm could be for instance built by the death and the consequent investigation from one or more characters, as in thrillers and spy stories, in which you can find a recursive plot, with multiple recursive steps even. In both these examples the syntagms can be considered as functions with a specific domain (the terms which compose them) and a co-domain equally specific, which is the code or the rhetorical meaning they refer to (manga comic or thriller story).
The inverse image of this function is contained in the domain of the original function, so in the case of the design, it selects the elements of the graphical interface of the document which can give the idea of a precise rhetorical signification; otherwise, in the case of the narration, it selects the events and characters to apply and in which order to do it to achieve a specific literary effect.
Consequently, if the function is made to define a sort of design you can say that the inverse image of it is the subset of all possible designs which returns that specific rhetorical meaning. In the case of the Japanese manga it could be composed by various elements (colors, typography, drawing style); these, however, constitute a subset of a more general set (which can be called “comic design elements”), used as the general domain of the original function.
For what concerns the narrative language, the comparison is not that easy. In order to see the inverse image of a function as a subset of its domain you need to consider all the events and the characters of a story, at a more abstract level than the one in which they are actually given to the reader. With this method you will be able to recognize that it is only a part of those events and characters the one which concurs to create the literary definition of the novel. The inverse image of the function will then be only a subset of the events belonging to the horizon of the novel.
Those are only some examples in which you can express mathematical contents.