Second Week BeyondResearch Insights, Year 1

Hello everybody! Welcome to the second week of my BeyondResearchInsights.

In the last session, we had an overview of relations in math. To put it briefly, relations are one of the most pervasive and important concepts in mathematics. That is, almost everything in math is a relation or deals with relations in some form.

An example that illustrates this point well is the membership or element relation. When we say that an element x is in a set S, then this is a relation. Set Theory is the most commonly used theory as the whole foundation of mathematics; thus, this relation is used everywhere.

The relation that is probably the most commonly used is equality. Indeed, this is also a relation. Equality is so important of a relation that, in most formulations of First Order Logic, they treat equality as part of the logic rather than an extra relation you can define. Equality inspires a whole class of relations called equivalence relations. These are kinds of relations that model equivalence, e.g. congruence of two triangles is an equivalence relation over triangles. Note how two congruent triangles aren’t necessarily the same triangle. As you can see there is a distinction between identity, or true equality, and equivalence. What I found interesting is that, apparently, homotopy-type theory (an alternative foundation for maths) looks at this distinction closer. Equivalence relations, similar to true equality, appear literally everywhere in math. When mathematicians say two things are “basically the same” (which happens a lot!) they are really talking about an equivalence relation, mostly an isomorphism.

Functions can also be modelled as special kinds of relations. Just think of defining a relation where you relate each element in the domain with its “output” in the codomain. Despite this, mathematicians don’t usually tend to think of functions as relations, that’s because they’re different concepts! They have different ideas behind them, it’s just that one can be modelled by the other. In fact, during the session, Dr. Bar (our mentor) alluded that you can model relations as functions as well! How do you do that? Well, that’s our job as scholars to figure out, and perhaps something interesting for you fellow readers to think about as well!

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