## Structures & Relations

Often spotting a *pattern* is the first step to understand the underlying *structure* involved.

And here lies the strength of *abstraction in mathematics*: the same mathematical structures can appear in widely different settings. Revealing a structure in one context can help solving problems in another context, no matter how different.

A *mathematical structure* consists of a collection of associated *objects* and *relation**s*,that must all satisfy certain requirements, called *axioms**.*

*Mathematical objects* are what we talk and write about when we do maths: numbers, functions, triangles, matrices, graphs, sets, etc. Despite their abstract nature, we think and talk about them as if they physically existed.

A *mathematical relation* is a relationship between sets of elements. Well-known relations in mathematics are *functions,* which associate to each element of a first set exactly one element of a second set.

In physics, many relationships are expressed by *equations* where quantities are represented by letters or symbols. *Algebra* is the branch of mathematics that studies these symbols and the rules for manipulating them.

*Axioms* are statements taken to be always true, to serve as a premise for further reasoning and arguments. They are fundamental building blocks of both *mathematical logic* and philosophy.

Combining those abstract concepts forms coherent systems, languages, and frameworks: *structures* allowing us to elaborate and communicate ideas, as well as to understand and model our world.