{"id":22796,"date":"2023-03-01T01:31:33","date_gmt":"2023-03-01T01:31:33","guid":{"rendered":"https:\/\/www.booksofall.com\/in\/?post_type=product&p=22796"},"modified":"2023-03-01T01:39:34","modified_gmt":"2023-03-01T01:39:34","slug":"the-zakon-series-on-mathematical-analysis-mathematical-analysis-volume-i","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/in\/the-zakon-series-on-mathematical-analysis-mathematical-analysis-volume-i\/","title":{"rendered":"The Zakon Series on Mathematical Analysis – Mathematical Analysis Volume I"},"content":{"rendered":"
\n

Chapter 1 –\u00a0Set Theory<\/h2>\n

1\u20133. Sets and Operations on Sets. Quantifiers<\/strong><\/p>\n

A set is a collection of objects of any specified kind. Sets<\/a> are usually denoted by capitals. The objects belonging to a set are called its elements or members. We write x \u2208 A if x is a member of A, and x\"\"A if it is not.<\/p>\n

A = {a, b, c, . . . } means that A consists of the elements<\/a> a, b, c, . . . . In particular, A = {a, b} consists of a and b; A = {p} consists of p alone. The empty or void set, \u2205, has no elements. Equality (=) means logical identity .<\/p>\n

If all members of A are also in B, we call A a subset of B (and B a superset of A), and write A \u2286<\/a> B or B \u2287 A. It is an axiom that the sets A and B are equal (A = B) if they have the same members, i.e., A \u2286 B and B \u2286 A.<\/p>\n

If, however, A \u2286 B but B \"\" A (i.e., B has some elements not in A), we call A a proper subset<\/a> of B and write A \u2282 B or B \u2283 A. \u201c\u2286\u201d is called the inclusion relation.<\/p>\n

Set equality<\/a> is not affected by the order in which elements appear. Thus {a, b} = {b, a}. Not so for ordered pairs (a, b).1 For such pairs,
\n(a, b) = (x, y) iff2 a = x and b = y,
\nbut not if a = y and b = x. Similarly, for ordered n-tuples,<\/p>\n

\"\"<\/p>\n

We write {x | P (x)} for \u201cthe set of all x satisfying the condition P (x).\u201d Similarly, {(x, y) | P (x, y)} is the set of all ordered pairs for which P (x, y) holds; {x \u2208 A | P (x)} is the set of those x in A for which P (x) is true.<\/p>\n

For any sets A and B, we define their union A \u222a B, intersection A \u2229 B, difference A\u2212B, and Cartesian product (or cross product) A\u00d7B, as follows: <\/span><\/p>\n

A \u222aB is the set of all members of A and B taken together :
\n<\/span>{x | x \u2208 A or x \u2208 B}.3 <\/span><\/p>\n

A \u2229B is the set of all common elements of A and B:
\n<\/span>{x \u2208 A | x \u2208 B}. <\/span><\/p>\n

A\u2212B consists of those x \u2208 A that are not in B:
\n<\/span>{x \u2208 A | x <\/span>\"\" B}. <\/span><\/p>\n

A\u00d7B is the set of all ordered pairs (x, y), with x \u2208 A and y \u2208 B:
\n<\/span>{(x, y) | x \u2208 A, y \u2208 B}. <\/span><\/p>\n

Similarly, A1\u00d7A2\u00d7\u00b7 \u00b7 \u00b7\u00d7An is the set of all ordered n-tuples (x1, . . . , xn) such that xk \u2208 Ak, k = 1, 2, . . . , n. We write An for A\u00d7A\u00d7 \u00b7 \u00b7 \u00b7 \u00d7A (n factors). <\/span><\/p>\n

A and B are said to be disjoint iff A \u2229 B = \u2205 (no common elements). <\/span>Otherwise, we say that A meets B (A \u2229 B<\/span>\"\"\u2205). Usually all sets involved are subsets of a \u201cmaster set\u201d S, called the space. Then we write \u2212X for S \u2212X, and call \u2212X the complement of X (in S). Various other notations are likewise in use. <\/span><\/p>\n

Examples.<\/span><\/p>\n

Let A = {1, 2, 3}, B = {2, 4}. Then
\nA \u222aB = {1, 2, 3, 4}, A \u2229B = {2}, A\u2212B = {1, 3},
\nA\u00d7B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}.<\/span><\/p>\n<\/div>\n

\n

If N is the set of all naturals (positive integers<\/a>), we could also write
\nA = {x \u2208 N | x < 4}.<\/p>\n

Theorem 1.<\/p>\n

(a) A \u222aA = A; A \u2229A = A;
\n(b) A \u222aB = B \u222aA, A \u2229B = B \u2229 A;
\n(c) (A \u222aB) \u222a C = A \u222a (B \u222a C); (A \u2229B) \u2229 C = A \u2229 (B \u2229 C);
\n(d) (A \u222aB) \u2229 C = (A \u2229 C) \u222a (B \u2229 C);
\n(e) (A \u2229B) \u222a C = (A \u222a C) \u2229 (B \u222a C).<\/p>\n

The proof of (d) is sketched in Problem 1. The rest is left to the reader.<\/p>\n

Because of (c), we may omit brackets in A\u222aB \u222aC and A\u2229B \u2229C; similarly for four or more sets. More generally, we may consider whole families of sets, i.e., collections of many (possibly infinitely many) sets. IfM is such a family, we define its union, \u22c3M, to be the set of all elements x, each belonging to at least one set of the family. The intersection of M, denoted \u22c2M, consists of those x that belong to all sets of the family simultaneously . Instead, we also write<\/p>\n

\"\"<\/p>\n

Often we can number the sets of a given family:<\/p>\n

A1, A2, . . . , An, . . . .<\/p>\n

More generally, we may denote all sets of a familyM by some letter (say, X) with indices i attached to it (the indices may, but need not , be numbers). The familyM then is denoted by {Xi} or {Xi | i \u2208 I}, where i is a variable index ranging over a suitable set I of indices (\u201cindex notation\u201d). In this case, the union and intersection ofM are denoted by such symbols as<\/p>\n

\"\"<\/p>\n

If the indices are integers, we may write m \u22c3<\/p>\n

\"\"<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"