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I've never taken a rigorous linear algebra class or a multivariable calculus or ODE/PDE class prior to studying cardinalities, injective,surjective, well-ordering principle, minimal counterexample, ZFC. I only know applied single variable calculus and linear algebra for scientists and engineers. Is the reason I'm not understanding how to prove that the real numbers are not countable and surjections and injections between sets and how to use the well ordering principle because I haven't seen how to use the implicit function theorem, how to do curl and divergence, and what's a non-homogeneous equation just to name a few topics from vectorcalculus/ODE/PDE?

Is the reason I'm bad at proving things about the cartesian product, division algorithim, injective and surjective on a characteristic function, powersets, unions and intersections, convex hulls, countability of rationals and integers, partitions of a set, event spaces infinite sequences product because I've never seen ODEs,PDEs, differential geometry?

Many of the people I know who are not struggling with proving things have taken multivariable calculus, vector calculus, ODE/PDE. I'm sure a genius would not struggle with writing proofs even if they never learned how to do ODE/PDE but for the average person, would not knowing ODE/PDE/Vector calculus mean that they haven't practiced or been taught the mathematical maturity needed to prove that the rationals are countable?

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    $\begingroup$ No, those are advanced topics and are not required to write proofs on basic topics. Nevertheless, learning to write proofs, like taking a class on real analysis, will make all subsequent math classes easier. $\endgroup$ – Michael Apr 15 '17 at 20:59
  • $\begingroup$ PS: I wrote up a description of the standard Cantor diagonal argument here, intended for the non-expert (no guarantee that this is any easier than other writeups, all writeups on this are doing essentially the same things): ee.usc.edu/stochastic-nets/docs/levels-of-infinity.pdf $\endgroup$ – Michael Apr 15 '17 at 21:03
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The answers to your specific questions about specific topics in mathematics are all technically "no." But that may be due merely to the way you phrased the questions.

You are correct that it helps to have more mathematical "maturity" when trying to develop proofs of such things as the uncountability of the real numbers. One gets that mathematical maturity by studying higher mathematics involving precise definitions and rigorous proofs.

But there is no reason why one could only achieve mathematical maturity via the specific topics you suggested. Given appropriate instruction, one could just as well gain mathematical maturity by studying other topics, even the topics you are studying now. Many people do it that way.

What you might want to do is to make your instructors aware that you are new to rigorous proofs, unlike many of your classmates. There may be things you can do during office hours that will help you catch up.

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