Subsets and subgroups in Lampião da Esquina
This week I read an excellent article written by my friend, historian Gabri Simionato, called “À margem da luz do lampião: travestis, bonecas e bichas loucas no Lampião da Esquina (1978-1981)“. It was a very interesting read, as it deals with a newspaper published between 1978 and 1981, a period in which Brazil was undergoing a military dictatorship, and several positions different from the “standard” by the dominant group were repressed through force. Until then it was not at all surprising to me that this conflict existed, and that they used violent methods to suppress the weakest link.
But Gabri Simionato raises a rather curious question, about the purposes for which a newspaper like Lampião na Esquina was “officially” published, how and why this happened, in addition to which aspects it “said” to serve and which it actually demonstrated to serve. It’s a very rich reading in details cut from readers’ letters, and from other historians who discussed this vehicle of communication, so I won’t give too many spoilers so as not to spoil the fun for those who still want to read :3
The coolest thing about this whole text (cool in the sense of surprising, although the event itself is tragic), is seeing how the conflicts between the parties weakened what we can see today as a “common cause”. In this case, for those who were “outsiders” and opposed to these causes, everyone in it was, so to speak, equal. But within the cause, the members themselves differed with some frequency, becoming hostile in apparent attempts to better position themselves in relation to others.
This text made me reflect that in addition to the social aspects, this situation also allows us to talk about two mathematical concepts usually confused in common language, Sets and Groups. Although in common language we do not apply much distinction between these two words, in mathematics these terms have different meanings:
- Sets are collections of elements
- Groups are sets of elements associated with an operation that combines any two elements to form a third.
Thus, every Group is also a Set, but not every Set is a Group.
Examples of sets are very simple: the set of objects above my table (notebook, thermos, cell phone, bible, honey, tea towel). I just realized that I need a cup and a spoon to drink tea, but these two elements are not part of this set.
Examples of groups are a little more difficult to formulate concretely, as we need to think of sets whose elements can be combined from a single action in order to generate another element of that set. An example of this could be the group of gelatin flavors (I already discussed a little about gelatins in the post 31 recipes with 5 gelatins), where from any combination of gelatin flavors, we will still be in the set of gelatin flavors. Unfortunately, that might involve the delicious grape-lemon mango jelly. Another example of a group, not so toxic to the stomach, are the colors (if we think of a system like RGB), any mixture of colors would still be within this same spectrum of variation.
An example of a set that is not a group are the objects on my table. It would be difficult to combine the thermos with honey and form the XD cell phone.
But how do these questions relate to the text about the newspaper Lampião na Esquina?
Now, let’s see how a difference in definitions has an impact on the whole context. To some degree it is impossible to deny that we are all part of the same set… if you want to be hardcore, you can classify yourself as an “aerobic organism”, that is, that needs oxygen, but that would put you on your feet on par with so many other living beings that you might prefer something a little less comprehensive. From there, the sets begin to funnel, in this case, we enter the subsets of aerobic organisms, that is, a set that is entirely contained within another set and another and another… the funnel continues, getting thinner and thinner. .. we don’t want to be just “human beings”, we want to be in the subset of human beings born in country X, or who exercise a certain profession, or who have a certain opinion… the funnel can be as narrow as we want, until we literally reach a subset in which only the person exists. Like the subset of people who write posts for this blog (hello darkness, my old friend).
It is already possible to see a little of how this discussion is associated with the article by Gabri Simionato, people who are in different subsets (for example, those who do not agree with a sexist and patriarchal system). But that within this subset, there are other subsets (for example, who does not agree with a sexist and patriarchal system for economic reasons and who does not agree for social reasons). Thus, there is fuel for an internal struggle between these subsets (one belittling the arguments of the other, for example, or even using their resources to reduce the cause of the other in preference to their own).
But where do subgroups come into this story? So while a subset is a set whose elements are inside the other… subgroups are more demanding entities. To begin with, every subgroup needs to be a group, but not every subset of a subgroup is a group. For example, Integers joined by the addition operation form a group. Because when we add any two elements of the Integers, we arrive at an Integer number.
But if we take a single number from this infinite set, for example, the Integers without the number 77. Although it is a subset of the Integers, it is not a subgroup of the Integers associated by the addition operation. See that if we take 76 and 1, the result will be 77, an element that no longer belongs to this subset.
However, the subset of integers without odds is a group with the addition operation. Because any two even numbers added together will be another even number.
The same does not happen if we think of the subset of Integers without the even numbers, because with the addition operation we will have any two odd numbers added together, it will be even, that is, it does not belong to the set, therefore it is not a group.
That said, we can see that subsets are very easy to form precisely because they carry few requirements, while subgroups are more difficult to form because they have more associations that need to be maintained.
In closing, we can make an analogy in Gabri Simionato’s discussion of Jornal Lampião na Esquina, which, despite the newspaper having the intention of satisfying the interests of a subset of society, had a very specific focus on a subset of this subset, chosen for reasons economic and social, making it non-representative and thus generating conflicts with those who claimed to represent the interests. If we think of the proposal as not only a collection, but also a preservation of characteristics and properties common to all, before the selection was made (as with the subgroups), when we narrow the set, we would be keeping these characteristics and properties.
In summary, the target audience of Jornal Lampião na Esquina was a subset of society “often oppressed by the social and political standards of the time” (common property of all in the group), but the way in which the newspaper directed its content to a subset of this subset, caused the characteristic common to them to be lost in part (that is, they no longer see themselves as belonging to the same group).