What is discrete mathematics based on?

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I am interested in studying computer science after I graduate, and I am aware that, based on your occupation, it can be a math heavy field. I've looking into discrete mathematics, and am interested in what the basis of it is. Also, if anyone has any experience, could you recommend any books for studying?

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Elementary logic, elementary set theory, formal arithmetic (developed from Peano axioms), some basic "structures": order, graphs, etc.