[...]...Mathematical and Physical Data, Equations, and Rules of Thumb Chapter 1 Algebra, Functions, Graphs, and Vectors 1 Copyright 2001 The McGraw-Hill Companies, Inc Click Here for Terms of Use This chapter contains data pertaining to sets, functions, arithmetic, real-number algebra, complex-number algebra, coordinate systems, graphs, and vector algebra Sets A set is a collection or group of definable unique... defined in terms of a sequence, even though there might be infinitely many elements in the set Some infinite number sets have non-denumerable cardinality; such a set cannot be completely defined in terms of a sequence One-one function Let A and B be two non-empty sets Suppose that for every member of A, a function f assigns some member of B Let a1 and a2 be members of A Let b1 and b2 be members of B, such that... assigns f(a1) ϭ b1 and f(a2) ϭ b2 Then f is a one-one function if and only if: a1 a2 → b1 b2 Onto function A function f from set A to set B is an onto function if and only if: b ʦ B → f(a) ϭ b for some a ʦ A 6 Chapter One One-to-one correspondence A function f from set A to set B is a one-to-one correspondence, also known as a bijection, if and only if f is both one-one and onto Domain and range Let f... integers, and natural numbers as follows: NʚZʚQʚR The operations addition, subtraction, multiplication, division, Algebra, Functions, Graphs, and Vectors 13 and exponentiation can be defined over the set of real numbers If # represents any one of these operations and x and y are elements of R with y 0, then: x#yʦR The symbol ℵ0 (aleph-null or aleph-nought) denotes the cardinality of the sets of natural... elements of non-denumerable number sets cannot be listed In fact, it is impossible to even define the elements of such a set by writing down a list or sequence Irrational numbers An irrational number is a number that cannot be expressed as the ratio of two integers Examples of irrational numbers include: Ⅲ The length of the diagonal of a square that is one unit on each edge Ⅲ The circumference-to-diameter... Continuum Hypothesis, the points on the so-called complex-number plane exist in a one-toone correspondence with the elements of C The set of imaginary numbers, J, is a proper subset of C The set of real numbers, R, is also a proper subset of C Formally: JʚC NʚZʚQʚRʚC 1.3 The imaginary numbers can be depicted as points on a vertical line Figure Algebra, Functions, Graphs, and Vectors 15 Figure 1.4 The complex... Algebra, Functions, Graphs, and Vectors 19 Figure 1.6 Calculation of absolute value (vector length) of a com- plex number a ϩ jb ϭ r cos ␪ ϩ j(r sin ␪) ϭ r(cos ␪ ϩ j sin ␪) The value of r, corresponding to the magnitude of the vector, is called the modulus The angle ␪, corresponding to the direction of the vector, is called the amplitude Product of complex numbers in polar form Let x1 and x2 be complex numbers... For all real numbers a1, a2, and a3, and for all complex numbers a1 ϩ jb1, a2 ϩ jb2, and a3 ϩ jb3, the following equations hold: (a1a2)a3 ϭ a1(a2a3) ((a1 ϩ jb1)(a2 ϩ jb2))(a3 ϩ jb3) ϭ (a1 ϩ jb1)((a2 ϩ jb2)(a3 ϩ jb3)) Distributivity of multiplication over addition For all real numbers a1, a2, and a3, and for all complex numbers a1 ϩ jb1, a2 ϩ jb2, and a3 ϩ jb3, the following equations hold: a1(a2 ϩ a3)... by A through F, the first six letters of the alphabet The digit set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F} Integers The set of natural numbers can be duplicated and inverted to form an identical, mirror-image set: Algebra, Functions, Graphs, and Vectors 9 ϪN ϭ {0, Ϫ1, Ϫ2, Ϫ3, , Ϫn, } The union of this set with the set of natural numbers produces the set of integers, commonly denoted Z: Z... numbers, integers, and rational numbers The cardinality of the real numbers is denoted ℵ1 (aleph-one) These ‘‘numbers’’ are called infinite cardinals or transfinite cardinals Around the year 1900, the German mathematician Georg Cantor proved that these two ‘‘numbers’’ are not the same: ℵ1 Ͼ ℵ0 This reflects the fact that the elements of N can be paired off one-to-one with the elements of Z or Q, but not .