Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 6

Set Theory

Set Theory, branch of mathematics, first given formal treatment by the German mathematician Georg Cantor in the 19th century. The set concept is one of the most basic in mathematics, even more primitive than the process of counting, and is found, explicitly or implicitly, in every area of pure and applied mathematics. Explicitly, the principles and terminology of sets are used to make mathematical statements more clear and precise and to clarify concepts such as the finite and the infinite.

A set is an aggregate, class, or collection of objects, which are called the elements of the set. In symbols, aS means that the element a belongs to or is contained in the set S, or that the set S contains the element a. A set S is defined if, given any object a, one and only one of these statements holds: aS or aS (that is, a is not contained in S). A set is frequently designated by the symbol S }, with the braces including the elements of = { S either by writing all of them in explicitly or by giving a formula, rule, or statement that describes all of them. Thus, S₁ = {2, 4}; S₂ = {2, 4, 6, ..., 2n,...} = {all positive even integers}; S₃ = {x | x² - 6x + 11 ≥ 3}; S₄ = {all living males named John}. In S₃ and S₄ it is implied that x is a number; S₃ is read as the set of all xs such that x² - 6x + 11 ≥ 3.

If every element of a set R also belongs to a set S,R is a subset of S, and S is a superset of R; in symbols, RS, or SR. A set is both a subset and a superset of itself. If RS, but at least one element in S is not in R,R is called a proper subset of S, and S is a proper superset of R; in symbols, RS,SR. If RS and SR, that is, if every element of one set is an element of the other, then R and S are the same, written R = S. Thus, in the examples cited above, S₁ is a proper subset of S₂.

If A and B are two subsets of a set S, the elements found in A or in B or in both form a subset of S called the union of A and B, written AB. The elements common to A and B form a subset of S called the intersection of A and B, written AB. If A and B have no elements in common, the intersection is empty; it is convenient, however, to think of the intersection as a set, designated by  and called the empty, or null, set. Thus, if A = {2, 4, 6}, B ={4, 6, 8, 10}, and C = {10, 14, 16, 26}, then AB = {2, 4, 6, 8, 10}, AC = {2, 4, 6, 10, 14, 16, 26}, AB = {4, 6}, AC = . The set of elements that are in A but not in B is called the difference between A and B, written A - B (sometimes A\B); thus, in the illustration above, A - B ={2}, B - A = {8, 10}. If A is a subset of a set l, the set of elements in l that are not in A, that is, l - A, is called the complement of A (with respect to l), written l - A = A’ (also written Ā,Ã, ~ A).

The following statements are basic consequences of the above definitions, with A,B,C,... representing subsets of a set l.

1. AB = BA.
2. AB = BA.
3. (AB) C = A (BC).
4. (AB) C = A (BC).
5. A = A.
6. A = .
7. Al = l.
8. Al = A.
9. A (BC) = (AB)  (AC).
10. A (BC) = (AB)  (AC).
11. AA’ = l.
12. AA’ = .
13. (AB)’ = A’B’.
14. (AB)’ = A’B’.
15. AA = AA = A.
16. (A’)’ = A.
17. A - B = AB’.
18. (A - B) - C = A - (BC).
19. If AB = , then (AB) - B = A.
20. A - (BC) = (A - B)  (A - C).

These are laws of the algebra of sets, which is an example of the algebraic system that mathematicians call Boolean algebra.

If S is a set, the set of all subsets of S is a new set D, sometimes called the derived set of S. Thus, if S = {a,b,c}; D ={{},{a}, {b},{c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Here,{} is used in place of the null set , of S; it is an element of D. If S has n elements, the derived set D has 2ⁿ elements. Larger and larger sets are obtained by taking the derived set D₂ of D, the derived set D₃ of D₂, and so on.

If A and B are two sets, the set of all possible ordered pairs of the form (a,b), with a in A and b in B, is called the Cartesian product of A and B, frequently written A × B. For example, if A ={1, 2}, B ={x,y,z}, then A × B ={ (1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}. B × A ={ (x, 1), (y, 1), (z, 1), (x, 2), (y, 2), (z, 2)}. Here, A × B≠B × A, because the pair (1, x) must be distinguished from the pair (x, 1).

The elements of the set A = {1, 2, 3} can be matched or paired with the elements of the set B = {x,y,z} in several (actually, six) ways such that each element of B is matched with an element of A, each element of A is matched with an element of B, and different elements of one set are matched with different elements of the other. For example, the elements may be matched (1, y), (2, z), (3, x). A matching of this type is called a one-to-one (1-1) correspondence between the elements of A and B. The elements of the set A = {1, 2, 3} cannot be put into a 1-1 correspondence with the elements of any one of its proper subsets and is therefore called a finite set or a set with finite cardinality. The elements of the set B = {1, 2, 3, ...} can be put into a 1-1 correspondence with the elements of its proper subset C ={3, 4, 5, ...} by matching, for example, n of B with n + 2 of C,n = 1, 2, 3, .... A set with this property is called an infinite set or a set of infinite cardinality. Two sets having elements that can be placed in a 1-1 correspondence are said to have the same cardinality.

Gauss, Carl Friedrich

Gauss, Carl Friedrich (1777-1855), German mathematician, noted for his wide-ranging contributions to physics, particularly the study of electromagnetism.

Born in Braunschweig on April 30, 1777, Gauss studied ancient languages in college, but at the age of 17 he became interested in mathematics and attempted a solution of the classical problem of constructing a regular heptagon, or seven-sided figure, with ruler and compass. He not only succeeded in proving this construction impossible, but went on to give methods of constructing figures with 17, 257, and 65,537 sides. In so doing he proved that the construction, with compass and ruler, of a regular polygon with an odd number of sides was possible only when the number of sides was a prime number of the series 3, 5, 17, 257, and 65,537 or was a multiple of two or more of these numbers. With this discovery he gave up his intention to study languages and turned to mathematics. He studied at the University of Göttingen from 1795 to 1798; for his doctoral thesis he submitted a proof that every algebraic equation has at least one root, or solution. This theorem, which had challenged mathematicians for centuries, is still called “the fundamental theorem of algebra” (see Algebra; Equations, Theory of). His volume on the theory of numbers, Disquisitiones Arithmeticae (Inquiries into Arithmetic, 1801), is a classic work in the field of mathematics.

Gauss next turned his attention to astronomy. A faint planetoid, Ceres, had been discovered in 1801; and because astronomers thought it was a planet, they observed it with great interest until losing sight of it. From the early observations Gauss calculated its exact position, so that it was easily rediscovered. He also worked out a new method for calculating the orbits of heavenly bodies. In 1807 Gauss was appointed professor of mathematics and director of the observatory at Göttingen, holding both positions until his death there on February 23, 1855.

Although Gauss made valuable contributions to both theoretical and practical astronomy, his principal work was in mathematics and mathematical physics. In theory of numbers, he developed the important prime-number theorem (see e). He was the first to develop a non-Euclidean geometry (see Geometry), but Gauss failed to publish these important findings because he wished to avoid publicity. In probability theory, he developed the important method of least squares and the fundamental laws of probability distribution (see Probability; Statistics). The normal probability graph is still called the Gaussian curve. He made geodetic surveys, and applied mathematics to geodesy (see Geophysics). With the German physicist Wilhelm Eduard Weber, Gauss did extensive research on magnetism. His applications of mathematics to both magnetism and electricity are among his most important works; the unit of intensity of magnetic fields is today called the gauss. He also carried out research in optics, particularly in systems of lenses. Scarcely a branch of mathematics or mathematical physics was untouched by Gauss.

Descartes, René

Descartes, René (1596-1650), French philosopher, scientist, and mathematician, sometimes called the father of modern philosophy.

Born in La Haye, Touraine (a region and former province of France), Descartes was the son of a minor nobleman and belonged to a family that had produced a number of learned men. At the age of eight he was enrolled in the Jesuit school of La Flèche in Anjou, where he remained for eight years. Besides the usual classical studies, Descartes received instruction in mathematics and in Scholastic philosophy, which attempted to use human reason to understand Christian doctrine (see Scholasticism). Roman Catholicism exerted a strong influence on Descartes throughout his life. Upon graduation from school, he studied law at the University of Poitiers, graduating in 1616. He never practiced law, however; in 1618 he entered the service of Prince Maurice of Nassau, leader of the United Provinces of the Netherlands, with the intention of following a military career. In succeeding years Descartes served in other armies, but his attention had already been attracted to the problems of mathematics and philosophy to which he was to devote the rest of his life. He made a pilgrimage to Italy from 1623 to 1624 and spent the years from 1624 to 1628 in France. While in France, Descartes devoted himself to the study of philosophy and also experimented in the science of optics. In 1628, having sold his properties in France, he moved to the Netherlands, where he spent most of the rest of his life. Descartes lived for varying periods in a number of different cities in the Netherlands, including Amsterdam, Deventer, Utrecht, and Leiden.

It was probably during the first years of his residence in the Netherlands that Descartes wrote his first major work, Essais philosophiques (Philosophical Essays), published in 1637. The work contained four parts: an essay on geometry, another on optics, a third on meteors, and Discours de la méthode (Discourse on Method), which described his philosophical speculations. This was followed by other philosophical works, among them Meditationes de Prima Philosophia (Meditations on First Philosophy, 1641; revised 1642) and Principia Philosophiae (The Principles of Philosophy, 1644). The latter volume was dedicated to Princess Elizabeth Stuart of Bohemia, who lived in the Netherlands and with whom Descartes had formed a deep friendship. In 1649 Descartes was invited to the court of Queen Christina of Sweden in Stockholm to give the queen instruction in philosophy. The rigors of the northern winter brought on the pneumonia that caused his death in 1650.

Descartes attempted to apply the rational inductive methods of science, and particularly of mathematics, to philosophy. Before his time, philosophy had been dominated by the method of Scholasticism, which was entirely based on comparing and contrasting the views of recognized authorities. Rejecting this method, Descartes stated, “In our search for the direct road to truth, we should busy ourselves with no object about which we cannot attain a certitude equal to that of the demonstration of arithmetic and geometry.” He therefore determined to hold nothing true until he had established grounds for believing it true. The single sure fact from which his investigations began was expressed by him in the famous words Cogito, ergo sum,”I think, therefore I am.” From this postulate that a clear consciousness of his thinking proved his own existence, he argued the existence of God. God, according to Descartes's philosophy, created two classes of substance that make up the whole of reality. One class was thinking substances, or minds, and the other was extended substances, or bodies.

Descartes's philosophy, sometimes called Cartesianism, carried him into elaborate and erroneous explanations of a number of physical phenomena. These explanations, however, had value, because he substituted a system of mechanical interpretations of physical phenomena for the vague spiritual concepts of most earlier writers. Although Descartes had at first been inclined to accept the Copernican theory of the universe with its concept of a system of spinning planets revolving around the sun, he abandoned this theory when it was pronounced heretical by the Roman Catholic church. In its place he devised a theory of vortices in which space was entirely filled with matter, in various states, whirling about the sun.

In the field of physiology, Descartes held that part of the blood was a subtle fluid, which he called animal spirits. The animal spirits, he believed, came into contact with thinking substances in the brain and flowed out along the channels of the nerves to animate the muscles and other parts of the body.

Descartes's study of optics led him to the independent discovery of the fundamental law of reflection: that the angle of incidence is equal to the angle of reflection. His essay on optics was the first published statement of this law. Descartes's treatment of light as a type of pressure in a solid medium paved the way for the undulatory theory of light.

The most notable contribution that Descartes made to mathematics was the systematization of analytic geometry (see Geometry: Analytic Geometry). He was the first mathematician to attempt to classify curves according to the types of equations that produce them. He also made contributions to the theory of equations. Descartes was the first to use the last letters of the alphabet to designate unknown quantities and the first letters to designate known ones. He also invented the method of indices (as in x²) to express the powers of numbers. In addition, he formulated the rule, which is known as Descartes's rule of signs, for finding the number of positive and negative roots for any algebraic equation.

Dyson, Freeman