A Transition to Advanced Mathematics
TRANSITION TO ADVANCED MATHEMATICS bridges the gap between calculus and advanced math in at least three ways. First, it guides students to think precisely and to express themselves mathematically—to analyze a situation, extract pertinent facts, and draw appropriate conclusions. Second, it provides a firm foundation of the basic concepts and methods needed for continued work. Finally, it provides introductions to concepts of modern algebra and analysis in sufficient depth to capture some of their spirit and characteristics. The text will improve the student's ability to think and write in a mature mathematical fashion and provide a solid understanding of the material most useful for advanced courses.
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A U B Abelian accumulation point algebraic system Assign a grade Assume Axiom of Choice bijection Bolzano-Weierstrass Theorem cardinal number choose Claim codomain commutative complex numbers contradiction contrapositive converges countable countable sets defined DEFINITION Let denumerable sets digraph direct proof divides domain equivalence classes equivalence relation Exercise exists F F F F failure F T F false Figure finite set function Give an example given by f(x graph Heine-Borel Theorem Hint implies induction infinite set integers inverse Let G mathematics multiplication natural number number of elements object one-to-one correspondence ordered field ordered pairs pairwise disjoint partial order partition positive integers positive real number prime Proofs to Grade Prove rational numbers real numbers reflexive Section symmetric tautology Theorem tion transitive truth table truth value vertex vertices