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Discrete Mathematics vs Mathematical Proofs: A Transition to Advanced Mathematics?

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At least in the United States, courses called "Discrete Math" or "Introduction to Proofs" are *incredibly* similar.  To the point that the Discrete Math course I teach actually transfers to some of the neighboring universities as their Intro to Proof.

So, there is almost certainly no reason to think you need to study one before the other.
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From the little I am able to gather, I would suggest starting out with the Discrete Mathematics text: it seems to be readily accessible to students.

See how this material suits you before progressing to more advanced material: you have to crawl before you walk, etc.
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Judging from the table of contents, the first book up to and including the chapter on algorithms looks like a good grounding. After that I suspect you would be better of with a work that's more geared towards computer science students than mathematicians. I am not sufficiently familiar with the subject to feel comfortable giving any recommendations on what to read, but *software verification* and *model checking* are search terms relevant to correctness of algorithms.

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