Note: While this question as phrased in the title is somewhat subjective, what I'm looking for as an answer should be specific enough to still have a clear/valid answer(s) to it. Also, apologies in advance if this is somewhat long winded but I feel the full context of my question is needed for clarity.
First let me provide some context. When it comes to learning new areas of math (or in this case, arguably, more fundamental/abstract areas), one problem all students face is 'vision'. Taking new concepts and adding them to old ones is often difficult since to do that requires seeing how things interrelate. However your typical education in math usually leaves you vastly unprepared to do this.
In a sense, most modern math education teaches students backwards. Most of the maths that students learn is gutted of the more important (and abstract) details. Details such as the 'real' proofs of certain theorems being replaced with a shorthand version that, while proves the part of the concept being focused on, ignores the parts that give it its foundation. "Proofs in a vacuum" as I call them since they usually don't bother proving (sometimes not even mentioning) other concepts that the main idea depends on.
This leads me to my problem. I have a rather strong foundation in multiple areas of math (particularly those often relevant to physics, such as calculus), but since my education has focused on results/applications rather then math as a whole I'm left without the 'glue' that ties it all together.
The language of set theory (as well as basic forms of logic) is still new to me. While I have a strong (enough) foundation in the basics after recently completing a class in discrete math, taking the concepts (and building onto them where needed) and applying them so as to add to my understanding of calculus for example is still difficult.
My question is this: What are some methods/pet projects I should try an implement to 'break out' of this novice phase as far set theory is concerned? To bridge the gap between knowing basic ideas concerning sets and relations between them and concepts from calculus for example? I don't want to look at an equation and see an equation. I wanna be able to see it as a more abstract entity.
Keep in mind what I'm looking for in an answer is:
- Specific topics to research that show how sets and relations are used to construct other others of math
- Go to examples for me to play around with so as to see how set theory is applied in practice (for example, how relations can be used to construct basic operations like addition, or differential operators).
- Examples of a common (but not obvious) mistakes people tend to make applying set theory to describe common concepts in other area of math.