A Bridge to Advanced Mathematics
This helpful workbook-style "bridge" book introduces students to the foundations of advanced mathematics, spanning the gap between a practically oriented calculus sequence and subsequent courses in algebra and analysis with a more theoretical slant.
Part 1 focuses on logic and number systems, providing the most basic tools, examples, and motivation for the manner, method, and concerns of higher mathematics. Part 2 covers sets, relations, functions, infinite sets, and mathematical proofs and reasoning.
Author Dennis Sentilles also discusses the history and development of mathematics as well as the reasons behind axiom systems and their uses. He assumes no prior knowledge of proofs or logic, and he takes an intuitive approach that builds into a formal development. Advanced undergraduate students of mathematics and engineering will find this volume an excellent source of instruction, reinforcement, and review.
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abstract algebra assumptions axiom of choice axiom system calculus called Cantor’s cardinal number chapter claim closed collection compact completeness axiom concept connectedness consider consisting continuous function contradiction correspondence countable countable set Deﬁne f deﬁned Deﬁnition denote denumerable discussion equation equivalence relation Euclidian Euclidian geometry example existence f is continuous ﬁnd ﬁrst ﬁxed function f geometry given idea implies induction inﬁnite sets integer intuitive Lemma Let f limit point logically equivalent mathe mathematicians matics matter meaning method natural numbers negation neighborhood non-empty notation number system one’s open interval open set parallel postulate point set precise problem propositional function prove quantiﬁed Question rational numbers reader real numbers reason relative topology result same-number of elements satisﬁed sense sequence set theory speciﬁc statement subset Suppose supremum symbol Theorem things tion topological space true truth-value uncountable upper bound usual topology words