On a Friday night, instead of going out and partying it up (if there is any partying in my system after all these final projects and exams) I decided to stay home and catch up on reading my intro to reasoning math books. I just went over the chapter that reviewed set theory and came across something interesting…. actually, something I forgot.

So in a set there are elements. Elements can be individual “items” (for lack of a better word) or it can also be another set. Sets can also be an element within another set. We can look at sets as boxes with things in it. These “things” (elements) can be one of the following three: nothing, something, or another box. You can have a box with nothing in it; this is a set with the zero element. However, when you put a thing in the box, you no longer have nothing in it but rather something in it! In addition, in your box (set, let’s call this box A), you can have a a thing in it (let’s say its a baseball), and you can have another box in it (let’s call this box B). We don’t care what is in box B, we just know that box B is an item inside box A. Now let’s say that in box B, there is a tennis ball.

If we ask, is one of the items in box A a tennis ball? You may be compelled to answer yes, but look at the question again. We’re asking for the items specifically in box A. In box A, there is a baseball and box B. There is no tennis ball in box A.

If we ask, does box A contain a tennis ball?  We can answer yes. Inside box A is box B and inside box B is a tennis ball.

Taking those ideas, we can express the following sets mathematically.

Sets are labeled with uppercase letters, elements are labeled with lowercase letters (or to avoid confusion we’ll use numerical values). Enumerating the items in a set are within { } (this is our box and what is between the brackets is what the box contains).

So let’s say set A = {2, 4, {1, 7}}
             where  B = {1, 7}
              then   A = {2, 4, B}

Let’s ask if one of the elements in A is 2. You see that in set A you have elements 2, 4, and B. So yes, 2 is an element of A.

Now is 7 an element in A? No. Recall, A has elements 2, 4, and B. There is no element that is 7.

… Okay, now I am sleepy. Peace out, y’all.