Radio Programs, Set Theory and Individuals



The Radio Broadcast Experience

I was listening to a National Public Radio program today, Talk of the Nation, and the hosts were discussing various topics with guests, and encouraging listeners to phone in or email them. The program's motif so to speak is based on allowing listeners to be heard by as the name implies--the whole country at large. But, as I listened to the callers that were put on air, an idea of how many listeners actually could be heard on this program began to form in my mind. The fact is very few listeners get the chance to speak to the nation. This fact is not very hard to see. Let’s take a fictitious example to illustrate it.

We have a radio called You might get on the air. It has 10,000 listeners. Now let’s see how many of that 10,000 can speak to the nation in one hour.

Radio You might get on the air =10000 listeners

Maximum number of listeners that can speak per minute if we assume the whole hour is devoted to listeners speaking is

10000/60=166 listeners per minute. This is a limit that the program will never reach. It is clear, the 166 people couldn’t be broadcasted in series speaking in one minute.


But course listeners aren’t given 1 minute to speak each, so let’s say we assume that each listener is given (at minimum) 10 seconds to speak. Ignoring access time, credits (even public radio has beginning and ending credits, though no commercials), we can see that if 10 seconds per call was the limit then we have:

10 sec *6=60 seconds which equals 1 minute, thus 6 callers speak per minute. 6 calls per minute * 60 = 360 callers get to speak on You might get on air.


The percentage becomes 360/10000=4% (approx) spoke on air, on the radio program You might get on the air.

These percentages are also the probabilities of your getting to speak. It is undeniably clear from this simple arithmetic that most of the people listening to You might get on air are never heard when they attempt to call in.  The fact is that most radio programs like this one that are nationwide have many more than 10,000 listeners. Also, note I'm telling the probability if the entire hour were devoted to listeners calling in. Of course, it's not. If we factor in the time the hosts take running their mouths, the probability is even less. You just have to count how many people you hear on air during a program like Talk of the Nation and do the arithmetic with an appropriate guess of how large the listening audience is, to see--very, I repeat very few are heard by the nation at large. My count is about 6 to 7 at most! I need to qualify these comments.  First, I am sure there are those who will cry no! no! no! He's wrong.  Suppose only 500 of the 10,000 attempt to call in, then the odds are much better?  I've thought of these objections too, and looked at what I project with differing numbers and probabilities, but still the percentage is small and another example illustrates. If only 500 of a possible audience size 10,000 attempts to call, and we assume that say maybe...uh 30 minutes of the 60 are given over to call-ins, then we have the probability of any one of the 30 getting to speak as 30/500=6% per minute. That's not a good chance again. The only way to increase the probability of a caller getting on air, is to increase the time allotted to call-ins, or a decrement in number of calls coming in. Both changes are unlikely in my opinion. But, on the rare occasions when the topic is uninteresting and there are not many calls, then more are heard. The greatest probability would be if only 30 callers phone in the 30 minutes, then every caller should be heard if we ignore the obvious constraints, access time, the screeners querying you before going on air, etc. But, that is not likely. I have heard on many PBS radio programs that they receive thousands of emails and calls. This is analogous to the numbers of callers phoning in during these programs. So, fiddling the numbers of my above estimation doesn't change the conclusion: most people are not heard on air.  The producers, hosts, and all others connected with these programs know this! For them to suggest their program as some kind of vehicle for you, the individual to be heard by the whole country is such a lie it makes me angry. And it’s even worse than the simple arithmetic implies, if you don’t play the game their way, even if you are chosen you won’t have the chance to speak to the country If you use profane language or expressing opinions they consider not appropriate you STILL won’t have your cherished chance to speak to their 50,000 or so listeners. It borders on false advertising to me. It is a lie that is foisted upon listeners to tell them to call in and let your voice be heard when the probability at most (drawing from the fictional example above) is 4% that they will be heard. Though I think its deceptive and unfair to make these false attributions to listeners, I am much more interested in what the radio broadcaster/listener experience implies about human relations. It should go without saying that whatever I write about radio as a media will apply to any other one-to-many relationship, such as TV, the Net, magazines, newspapers. We will see using database theory terms and set theory methods, that individuals have a special relationship with the multitude.

The Individual and The Multitude

As individuals, we are anonymous to the broadcasting world of radio. Of course, they have all sorts of sophisticated methods of knowing us in general. They typify us by race, age, stations we listen to, income, geography, consumption patterns, sex etc. Yet even with these data the broadcasters (I mean the entire organization by that term) don’t know us as individuals. That is to say, when we tune in a program, there is no person or persons there to see and say… oh that’s X, who is now lighting a cigarette, and okay he told his wife to be quiet, and uh he’s going to take a uh let me see… oh yeah he’s taking a piss before we start broadcasting that's old X, he always takes his piss before listening to us.  Thank the non-existent God they don't have that power yet, huh? There is a good reason why they don’t have that kind of power yet: they couldn’t handle the information flow that level of detail would require, even with powerful computer monitoring technology maybe in 50 years.  No, we are for them the amorphous, anonymous audience out there. We are the Many dialing into the One. The One has the Godlike power to reach the multitude, while the Many is defined by being a supplicating hoard seeking to speak to the One. Yes if you noticed, this relationship is the same one that is characteristic of religious experiences. When the Many at last finds and can talk to the One, what happens then? This is equivalent in the radio example to a caller having his chance to talk to the nation.

If you’ve ever had the experience of being chosen to give your opinion on a call-in radio show, you’ll no doubt know that it’s an overwhelming one.  When you hear your name announced on some radio program from such and such place, and then the hosts ask you to speak, you are at first shocked to hear your NAME called out and knowing you will be speaking to tens of thousands of people, you are nervous and apprehensive. If you pull it off, you feel a sense of worth and accomplishment. What has happened in these cases is the many has become a part of the one. You, the individual for those brief seconds, become like the radio show host, touching the multitude out there. You feel something akin to stage fright, your voice may crack, and you might ramble, and completely forget what you had composed in your head, that important point you were going to make, it now seems so small. Or contrarily, you may take control of yourself and speak clearly, and put your point across eloquently. In either case, you feel as if you’ve been given a power: the power to communicate with some many, many others. And you also sense something comes with this. You lose your anonymity in doing this. You are not a private person any longer. This is disturbing to you. And why is that?

We are, all of us private beings. We share the knowledge of our private states of mind, by choice. We think in our heads and have experiences only known to us. Privacy is a part of our existence we have from birth. The child sliding out of its mother’s womb is a child experiencing that occurrence alone. That same child while in the womb, if it has a rudimentary thought process anything like it will have as an adult it's still doing this alone. Even its mother carrying it doesn’t know that it might be developing thoughts. So you see, we are by our very nature as living beings in this world alone. And to be alone means to be private. Not even a fundamental bond like that of the pregnant mother to her infant can break that necessary divide of being a living thing alone and private unto itself. Is it any wonder that when we are called upon to share ourselves through something like a radio broadcast, we feel a cringing in ourselves?

Then there is this need we feel to communicate with others.  We are unto ourselves a unified agency of states of mind. Still, we do want to know others and communicate with them. I would venture to say, the majority of us want to touch many others too. We want to be heard, by the many if you will as the one. Most of the time, we only want to communicate in as a one-on-one relationship. Again another term from database theory, I’m using. We meet and know others as individuals like ourselves, and are themselves alone as human beings in this world. So what would ever make us want to know more than our individuated experiences can offer? It is the social nature of our being in the world that does this.

We are a mix of several types. Humanity is not one individual or type of individual. We are in our genetic combinations so many types (members) of one set. The human set, which is outward and forming new relationships, and thus forming new sets from its generating set, is a process that defines our social world. This behavior in human beings makes me think we are somehow doing this from something more basic. Let’s see if we can build a model of human communications as a relationship of sets, and apply it to the radio broadcast example above.

 Modeling The Radio Broadcast Experience

If we take the set of integers, {0, 1,2, 3∞}, then the only element of that set that under the operation of + has relation to every other member of that set is 0. And every element of the integer set has a one-to-many relationship with the number 0 called Identity. So, 0 is like the broadcaster above, and we are like the other integers. The problem here is when we call 0, we get ourselves as the result. This is a very simple set with rules that map in a way that the broadcaster never let’s the members in the set talk to anyone but themselves. Not a very fruitful example. So, let’s consider a set relation that widens the field and somewhat approximates the radio call-in experience. Logarithms of base 10 can capture this idea.

Consider the series below:

It is clear if we keep going on a set integers would be produced from increasing powers of the logarithms of these base 10 numbers.  Now go back to my original example. If we consider the callers as the result of a logarithmic set, then we have the following equation.


F(x) =xlog10

This set would generate every integer to infinity for powers of 10. What does this mean? It means in simple terms, smaller numbers would be mapped to larger numbers. This mapping approximates the one-to-many nature of radio broadcasting.

For example, take 4 log 10 = 10000. We could see this as indicating 4 people (the host and a small staff) communicate with 10,000. In this manner, the radio broadcast experience can be said to have a logarithmic relation. Though, to be more accurate about it, we’d probably have to change the base of the logarithm. But, there are other ways to capture the relationship between the broadcaster and the audience. A few examples will get us started.  Let's go through some of them.


Take the equation


F(x) =√x for x≠1 and x is an integer

This relationship is a function for all integers greater than 1. In other words, it is a one-to-one mapping. This relationship is said to be isomorphic, since every input begets a unique output within the set of integers. This relationship is more like a conversation between individuals than a broadcast. Though one person starts all the communicating, that is x starts conversations.

Composite functions also approximate one-to-one mappings, though less uniformly.


Take the equations


F(x) = 2x +1/x2

G(x) =2(f(x)) +1=2(2x + 1/x2) + 1= 4x +2/ x2 + 1= 2/x2 + 4x + 1


Here for every input to F(x) we get a unique output in G(x), often called the image of F(x). These sets are like the above isomorphic mapping and would be another person-to-person sort of communication. However, a subclass of composition known as iteration is very much like an exchange in which one speaks and the other responds using the information that was given from the original speaker.


Take the iterative equation

F(x) =

F(F(x)) for F(x)= 1/√x-1. It is a real-valued function beyond 0 and 1

As a non-iterated function, this set approaches from the left (that is, decreasing number values for x) the value of 0 as shown below:


F(x) = 1/√x-1 = 0

Lim x->∞ 

Which means conversation dies off between the two mapped sets. It would be like a one-to-one mapping, where one side stops communicating.


F(x) = 1/√x-1 = 0 and

F((F(x))= 0 also.



Embedding this function in itself and taking its limit leads to again a slow slide to 0. It will take longer no doubt, but the conservation eventually dies off.

 Real Communication: The Individual Meets the Multitude


None of the above, set mappings captures what happens in the radio broadcast I started this article with. But there is a way to make the exchange between the broadcaster and audience, more symmetric.


We now come back to the one-to-many set mapping we started with using base 10 logs. But instead of base 10 logs we will use base 2 logs. This set relation provides a much more realistic model of broadcasting to a wide audience, for instance, look at this:


F(x) = 20 log2 = 1,048,576.


Now that is much closer to the kind of relationship a show like Talk of the Nation has with its listeners. This mapping is saying that a staff of 20 can reach 1,048,576, but they don’t talk back much. Here is what I have been leading up to: why not let groups of listeners form sets that can talk back to the broadcaster as a group. The broadcasters will still decide what listeners will have the chance to speak, but the basis for this decision has more equanimity and would be representative of the audience. This model can be made with set theory methods. It would be cost-effective in the economic sense of that term. You could dispense with the jerks screening the calls with this model.


We can use the base 2 log above to develop a model that would allow callers to radio programs like Talk of the Nation to voice their opinions in large numbers. The model utilizes the database theory idea of one-to-many, but its converse: many-to-one.  for my model to be realized, the radio show's producers would have to do more preparation to accommodate their mass of callers. It would take an entire week before the broadcast airs for the show's producers to set up what my model illustrates. This isn't asking too much of them. After all, this program is the Talk of the Nation, and it should strive to be just that, right?

Go on to next section Sets, Radio programs and Individuals