So far, we’ve presented the (3) systems’ axioms and the notions of system’s behavior and system’s boundary. We have also explored these ideas via different examples. And we’ve touched on the idea of a set. However, we now want to differentiate between what a set is and what a system is. Once we show the difference between the two, then we will be able to demonstrate the difference between a subset and a subsystem. And most importantly, we will be able to better observe, analyze, and make sense of different kinds of systems, albeit economic systems, political systems, or political systems.

So how can we differentiate between a set and a system? First, we can address this question by referencing back to the (3) systems’ axioms:

- A system consists of a set of elements.
- Elements in a system interact.
- A system has a function, or purpose.

The difference between a set and a system is that a set satisfies the first axiom; whereas, a system satisfies all three axioms. More specifically, a set B is a collection of well-defined objects (we will use Naive Set Theory for now), for instance B = {2,4,6,8,10}. Further more, the elements in this set interact with each other. For example, the element ‘2’ interacts with element ‘8,’ or element ‘4’ interacts with element ’10,’ or some combination of possible interaction. And finally, there is a function that is produced, or purpose, via the interactions.

As we can see, a set satisfies the first axiom; whereas, a system satisfies all three axioms. Now we have the tools to delve into the subsets and subsystems. We will see that subsets satisfy the first axiom while subsystems satisfy all three axioms.

As stated before, a set B is a collection of well-defined objects, for instance B = {2,4,6,8,10}. However, a subset of B can be partitioned and observed. For instance, a subset A is a subset of set B if all of the elements in the set A are contained in the set B. That is, A = {2,4,6} so since all of the elements in the set A are contained in the set B, the set A = {2,4,6} is a subset of set B = {2,4,6,8,10}.

Thus, this subset or any combination of subsets with any of the five elements – 2,4,6,8,10 – satisfies the first system’s axiom.

To illustrate the second axiom with respect to a subsystem, we want to show that if elements interact in a subsystem, then they interact in a parent system. There are a few ways we can do this. For this article, we can do this by observing the interactions in set A = {2,4,6}. Thus if ‘2’ interacts with ‘4’ and ‘6,’ and ‘4’ interacts with ‘6’ in set A, then these elements also interact in set B because set A = {2, 4, 6} is a subset of set B = {2, 4, 6, 8, 10} because set A is contained in set B.

The final step is to show that a subsystem has a function, or purpose. It could be the case that a subsystem has the same function as its parent system, or it could be the case that it has a function different from its parent system. But either way, it ought to have a function no matter if it is the same or different from its parent system. So how can this be illustrated?

As Donella Meadows conveyed in her book *Thinking in Systems: A Primer* identifying the function of a system can sometimes be difficult. Indeed, there are instances where the function or a system is fairly obvious.

One way this can be done is by mapping the elements in set B to the elements in set A. In other words, the elements in set B will go to the elements in set A.

The sketch in *Example 1* illustrates this point. For instance, 1 goes to 3, and 2 also goes to 3; 4 goes to 7; and 5 goes to 8.

And so something is imputed through 1, 2, 4, and 5, and something is outputted through 3, 7, and 8. This means the elements in set B = {1, 2, 4, 5} would be the inputs of the system and the elements in set A = {3, 7, 8} would be the outputs.

To illustrate this point further, one could view a system that includes labor and wages as the elements. That is, a person exchanges their labor, hours worked, for a wage. If, for example, the wage was set at $30 per hour, then a person would obviously make more for every hour worked as *Graph 1* shows.

That is, if 5 hours are imputed into the system, then $150 will be outputted from the system; if 6 hours are imputed into the system, then $180 will be outputted from the system; and if 7 hours are imputed into the system, then $210 will be outputted from the system. And of course this game could be played over and over again. Thus, as the number of hours imputed into the system increases, the number of dollars outputted from the system increases.

Another demonstration of a function can be illustrated through an interaction between an oxygen molecule, O2, and two hydrogen molecules, 2H2. If a gaseous oxygen molecule interacts with two gaseous hydrogen molecules at a high temperature, these molecules are known as the reactants in chemistry, then two gaseous H2O molecules, known as the products in chemistry, will be produced. In other words, if one gaseous oxygen molecule and a two gaseous hydrogen molecules are imputed into a system, then the system will output two gaseous H2O molecules as *Example 2* demonstrates.

These systems’ functions and purposes are obviously not what we often think of as a function or purpose of a system. They are in one instance somewhat familiar and in another instance esoteric.

In this article, we have used mathematics along with a couple of examples from economics and chemistry to distinguish the difference between a set and a system. Moving forward, we will be able to continue building off of these axioms, notions, and examples as we begin to apply these ideas to more familiar systems such as economic systems, political systems, and social systems.

Let us now, as we have done before, attempt to disprove our notions and work in the tradition of natural philosophy until the next blog.

*Matt Johnson is a blogger/writer for The Systems Scientist and the Urban Dynamics blog. He has also contributed to the Iowa State Daily and Our Black News. *

*Matt** has a Bachelor of Science in Systems Science, with focuses in applied mathematics and economic systems, from Iowa State University. He is also a professional member of the Society of Industrial and Applied Mathematics and the International Society for the Systems Sciences and a scholarly member of Omicron Delta Epsilon, which is an International Honors Society for Economics. *

*Y**ou can connect with him directly in the comments section, and follow him on Facebook. *

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