Sets, Statements, and Variables: Introduction; Sets and their members; Construction of sets; Variables and statement-forms; Functions; Rules of inference and proofs; Beliefs, validity, and sets; Exercises Cardinal Numbers: Introduction; Standard sets and cardinal numbers; Addition of cardinal numbers; The commutative and associative laws of addition; Multiplication of cardinal numbers; The commutative and associative laws of multiplication; The distributive law; Comments on the five basic laws; Cancellation laws; Special properties of zero and one; The addition and multiplication tables for cardinals; Exercises Expressions: Introduction; Parentheses; Expressions; Evaluation of expressions; Equal expressions; The algebra of expressions; Parentheses and the generalized addition law; Parentheses and the generalized multiplication law; Parentheses and the generalized distributive law; Exercises Polynomials: Introduction; Expressions with no parentheses; Exponents; Monomials; Multiplication and addition of monomials; Polynomials; Exercises Number Systems in General: Introduction; Operations on a set; Commutative and associative operations; Distributive operations; Numbers and number systems; Zero-element and one-element of a number system; Inverses, negatives, and reciprocals; Some remarks about numbers in general; Exercises Construction of the Integers: Introduction; Definition of integers; Relations among the sets $I (x, y)$; Definition of addition for integers; The commutative and associative laws for addition of integers; Definition of multiplication for integers; The commutative and associative laws for multiplication of integers; The distributive laws for integers; Relation between cardinals and integers; Isomorphisms; Exercises Properties of the Integers: Introduction; The generalized laws and expressions; The zero-element and one-element of the integers; The negative of an integer; Standard notation for the integers; The cancellation laws for integers; Expressions over the integers; Subtraction; Exercises The Rational Number System: Introduction; The set of rational numbers; Addition and multiplication of rationals; One-element, zero-element and negatives; Relation between rationals and integers; Reciprocals; Traditional notation for rationals; Standard forms; Expressions over the rationals; Exercises Equations: Introduction; Definitions; Classification and solution of equations; Equations in one variable; Equations of the first degree; Quadratic equations; Equations with more than one variable; Simultaneous equations; Additional comments; Exercises Order: Introduction; Order for the cardinals; Order for the integers; Order in general; Order for the rationals; Exercises The Real Number System: Introduction; Finite decimals; The rationals as infinite decimals; Repeating decimals; Construction of the real number system; Order properties of the real number system; The reals as an extension of the rationals; Extraction of roots; Exponents; Logarithms; Exercises Appendix: Introduction; Infinite sets and the cancellation laws; Equal and unequal infinities; Peano axioms for natural numbers; Proof by mathematical induction; Axioms, groups, rings, and fields; Exercises Index.
