Know what a fully formal proof looks like. However, you are never. Expected to write down all of your arguments in that level of detail. Unless you Funny Black Girl Pittsburgh Steelers Girl Shirt proof assistant like Coq or Agda). All that is expected of you is, that if you write a non-fully detailed proof. And if someone asks you how a particular step works. That you then can provide a more detailed justification for that step. Mainstream math is often in the form of tools (like dividing both sides of an equation by some number). These tools are ingrained in our heads by working out exercise after exercise. They are also somewhat ‘intuitive’: maybe the first time you learned. To divide both sides by some number was very magical. But I think at least after some time, you felt like you could simply accept it.
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Think about students in the early stages of education asking ‘why can we do this?’ and the teacher trying their best to explain some ‘intuitive’ reason or justification or even ending with ‘these are the rules’ if nothing else works. For a lot of people, the tools are enough to get them Funny Black Girl Pittsburgh Steelers Girl Shirt. The first thing one has to do when starting to study maths is realized. That a lot if not all of the tools we have been taught are not justified. As such, the first math course you will usually take is about some stuff you are familiar with, but with a strong emphasis on justification/proofs and how to write them. You will probably have some intuition to guide you, and there can just be a lot of emphasis on learning.
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