Equality relation – Arkadiusz Jadczyk blog

I learn that:

“In the school year 2022/2023 they will have a new subject: history and the present. In-depth reforms will also take place in teaching education for security. These classes will be enriched with elements of defense training (including shooting, cybersecurity, field studies).”

The “present” has been added to the story. I wonder what the textbooks of the present will look like. There will be no textbooks of the future, because the future is highly doubtful – the world is moving faster and faster towards self-destruction. We have a big problem with the present. As my wise friend aptly put it:

“The solution to the problems lies outside the realm in which the problems were made. Outside the 3D realm.”

What is this 3D? This cannot be explained without mathematics. And you have to learn math from scratch. From set theory. However, this one also offers pitfalls. Set theory is the science of sets. A thinking high school student, when he finds out what these “sets” are, will ask: is there a subset of the set of all sets composed of those sets that are not elements of themselves? (Russell’s Paradox). And the colleague from the bench will think for a moment and simplify this question to the question “is there a collection of all collections at all?”

To be able to answer similar questions, we need to learn something about relations and the equality of sets. And today we will deal with this using the example of angel’s wings.

From “The Theory of Set”, Kuratowski-Mostowski:

Definition: Two sets A and B are equal if there exists a reciprocal function f for which A is a set of arguments and B is a set of values. We then write A∼B. About the function f we say that it establishes the equivalence of sets A and B.

That’s the definition. By definition, it is easy to see that the sets A = {a, b} and B = {1,2} are equal. We can easily construct the function f. Even two different. (Do it!)

It is more difficult to prove that sets A = {a, b, c} and B = {1,2} they are not equal. Someone will say: “they are not because one has three elements and the other only two”. However, this will not be proof by definition. It would be evidence from a certain conclusion by definition, but that conclusion will come after that.

In practice, the harvest often turns out fuzzy. Let’s take a look at a girl with angel wings.

Treating the wings as a collection of feathers, we can check their parallelism by drawing the following f:

However, when we come to the inner folds, a problem arises: should I count them for feathers? or not? It arose from such considerations fuzzy logic i fuzzy set theory. We then add weight to say that something is a pen, a number between 0 and 1. But this is different.

What is important to us is that the equality relation is an equivalence relation: reflexive, symmetric and transitive:


(A∼B) →(B∼A)


In this way, we introduce equivalence classes in a non-existent set of all sets. In a given class all sets are equal, and in different classes they are not equal.

And so we come back to the problem: prove, by definition, that sets A = {a, b, c} and B = {1,2} do not belong to the same class.

When we prove it properly, then we will be able to tackle slightly more difficult problems, such as “how to construct a time machine?”

PS Interestingly, my note today 05-09-22 at 16:04 went to the main page of salon24 with a puzzling illustration added by admins:


Source: Salon24.pl: Strona główna by www.salon24.pl.

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