[...]... Z and Z+ have the same cardinality We denote this cardinal number by �o, so that IZI = I Z+ I = �o It is fascinating that a proper subset of an infinite set may have the same number of elements as the whole set; an infinite set can be defined as a set having this property We naturally wonder whether all infinite sets have the same cardinality as the set Z A set has cardinality �o if and only if all... For any set A, finite or infinite, let B A be the set of all functions mapping A into the set B = {O, I } Show that the cardinality of B A is the same as the cardinality of the set 3" (A) [Hint: Each element of B A determines a subset of A in a natural way.] 19 Show that the power set of a set A, finite or infinite, has too many elements to be able to be put in a one-to-one correspondence with A Explain... you a little idea of the nature of abstract algebra We are all familiar with addition and multiplication of real numbers Both addition and multiplication combine two numbers to obtain one number For example, addition combines 2 and 3 to obtain 5 We consider addition and multiplication to be binary operations In this text, we abstract this notion, and examine sets in which we have one or more binary... Numbers A real number can be visualized geometrically as a point on a line that we often regard as an x-axis A complex number can be regarded as a point in the Euclidean plane, as shown in Fig 1 1 Note that we label the vertical axis as the yi -axis rather than just the y-axis, Cartesian coordinates (a, b) is labeled a + bi in Fig 1 1 The set C of complex numbers is defined by C = {a + bi I a b E lR}... correspondence satisfying the relation (8) is called an isomOlphism Names of elements and names of binary operations are not important in abstract algebra; we are interested in algebraic Section 1 Introduction and Examples 17 properties We illustrate what we mean by saying that the algebraic properties of U and of � 2J[ are the same 1.11 Example In U there is exactly one element e such that e Z = z for all z... B It seems reasonable to expect a one-to-one mapping simply to be a mapping that carries one point of A into one point of B, in the direction indicated by the arrow But of course, evel}' mapping of A into B does this, and Definition 0 1 2 did not say that at all With this unfortunate situation in mind, make as good a pedagogical case as you can for calling the functions described in Definition 0 1... Z and Z + have the same cardinality Such a pairing, showing that sets X and Y have the same cardinality, is a special type of relation++ between X and Y called a one-to-one correspondence Since each element x of X appears precisely once in this relation, we can regard this one-to-one correspondence as afunction with domain X The range of the function is Y because each y in Y also appears in some pairing... binary operations We think of a binary operation on a set as giving an algebra on the set, and we are interested in the structural properties of that algebra To illustrate what we mean by a structural property with our familiar set lR of real numbers, note that the equation x + x = a has a solution x in lR for each a E lR., namely, x = a1 2 However, the corresponding multiplicative equation x x = a does... that multiplication of n x n matrices is an associative binary operation If, in a linear algebra course, we first show that there is a one-to-one correspondence between matrices and linear transformations and that multiplication of matrices corresponds to the composition of the linear transformations (functions), we obtain this associativity at once from Theorem 2 1 3 Tables For a finite set, a binary... be listed in an infinite row, so that we could "number them" using Z+ Figure 0 15 indicates that this is possible for the set Q The square array of fractions extends infinitely to the right and infinitely downward, and contains all members of Q We have shown a string winding its way through this array Imagine the fractions to be glued to this string Taking the beginning of the string and pulling to the .   A      A     A  .              I A I   .     A   