Introductory Concepts for Abstract Mathematics
Beyond calculus, the world of mathematics grows increasingly abstract and places new and challenging demands on those venturing into that realm. As the focus of calculus instruction has become increasingly computational, it leaves many students ill prepared for more advanced work that requires the ability to understand and construct proofs.
Introductory Concepts for Abstract Mathematics helps readers bridge that gap. It teaches them to work with abstract ideas and develop a facility with definitions, theorems, and proofs. They learn logical principles, and to justify arguments not by what seems right, but by strict adherence to principles of logic and proven mathematical assertions - and they learn to write clearly in the language of mathematics
The author achieves these goals through a methodical treatment of set theory, relations and functions, and number systems, from the natural to the real. He introduces topics not usually addressed at this level, including the remarkable concepts of infinite sets and transfinite cardinal numbers
Introductory Concepts for Abstract Mathematics takes readers into the world beyond calculus and ensures their voyage to that world is successful. It imparts a feeling for the beauty of mathematics and its internal harmony, and inspires an eagerness and increased enthusiasm for moving forward in the study of mathematics.
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Logic and Propositional Calculus
Tautologies and Validity
Quantifiers and Predicates
Techniques of Derivation and Rules
Sets and Set Operations
Set Union Intersection and Complement
Composition of Functions
Image and Preimage Functions
ALGEBRAIC AND ORDER PROPERTIES
Finite and Infinite Sets
Denumerable and Countable Sets
Transfinite Cardinal Numbers
Partially Ordered Sets
A C B A U B addition and multiplication algebraic argument assertion axiom binary operation cardinally equivalent chain Chapter commutative compound statement conjecture Consequently considered contrapositive corollary countable defined definition denoted derivation method direct proof disjoint equation equivalence relation established example Exercise F F F F T F false finite sets function given Hence implies infinite sets integers intersection inverse least upper bound lemma lower bound mathematical induction maximal element natural numbers negation nonempty set order isomorphism ordered pairs P A Q P V Q partial order relation partially ordered set partition poset predicate preimage premises properties Prove Theorem quantifiers real numbers reflexive respect result Rule E.S. Rule U.G. says set builder notation set theory step student Suppose symbol symmetric tautology transitive trichotomy law true truth table truth value variable Verify Vx[x words