[...]... element its logarithm (to base e) since logee" = x and e 1og." = x The inverse of the product of two or more mappings or transformations (provided they are both one -to- one) can easily be found For suppose f sends x into y and 9 sends y into z so that y = f(x) and z = g(y) 6 (9) Sets, Mappings and Transformations [1.4) Then z = g[J(x)], (10) which, by definition, means first perform f on x and then g on f(x)... AN INTRODUCTION TO MATRICES, SETS AND GROUPS FOR SCIENCE STUDENTS CHAPTER 1 Sets, Mappings and Transformations 1.1 Introduction The concept of a set of objects is one of the most fundamental in mathematics, and set theory along with mathematical logic may properly be said to lie at the very foundations of mathematics Although it is not the purpose of this book to delve into the fundamental... of mn quantities arranged in m rows and n columns is called a matrix of order (m x n) and must be thought of as operating on X in such a way as to reproduce the right-hand side of (20) The quantities a 1k are called the elements of the matrix A, alk being the element in the i 1h row and k 1h column We now see that 9 Sets, Mappings and Transformations [1.5J Y and X are matrices of order (m xl) and (n... set (y Y2' , Ym)' An inverse transformation to (20) cannot exist therefore, and consequently we should not expect to be able to find an inverse matrix A -1 (say) which undoes the work of A Indeed, inverses of non-square matrices are not defined However, if m = n it may be possible to find an inverse transformation and an associated inverse matrix Consider, for example, the transformation (19) Solving... theory and its notation to enable the ideas of mappings and transformations (linear, in particular) to be understood Linear transformations are then used as a means of introducing matrices, the more formal approach to matrix algebra and matrix calculus being dealt with in the following chapters In the later sections of this chapter we again return to set theory, giving a brief account of set algebra together... pursuing as it leads naturally on the one hand into such concepts as mappings and transformations from which the matrix idea follows and, on the other, into group theory with its ever growing applications in the physical sciences Furthermore, sets and mathematical logic are now basic to much of the design of computers and electrical circuits, as well as to the axiomatic formulation of probability theory In... However, even two one -to- one mappings do not necessarily commute Nevertheless, two mappings which always commute are a one -to- one mapping f and its inverse f- 1 (see (8) ) 1.5 Linear transformations and matrices Consider now the two-dimensional problem of the rotation of rectangular Cartesian axes x 10X Z through an angle e into Y10yZ (see Fig 1.6) .p o xJ Fig 1.6 8 Sets, Mappings and Transformations [1.5]... little farther here A common criticism of introducing set theory to scientists and engineers (and for that matter to school children, as is now fashion14 Sets, Mappings and Transformations (1.7) able) is that it is only notation and that little or nothing can be done using set formalism that cannot be done in a more conventional way Although to some extent this may be true it is equally true of a large... of a new notation often has a unifying and simplifying effect and suggests lines of further development For example, it is more convenient to deal with the Arabic numbers rather than the clumsy Roman form; vectors are more convenient in many cases than dealing separately with their components, and linear transformations are better dealt with in matrix form than by writing down a set of n linear equations... U]j represented by shaded region Fig 1.8 Clearly AuB= BuA, (46) and, since A and B are subsets of A u B, A £; (A u B), and B £; (A u B) (47) Likewise Au U = UuA = U 15 (48) Sets, Mappings and Transformations (1.71 The intersection (or meet or logical product) of A and B is denoted by A (') B and is the set of those elements common to both A and B (see the shaded part of the Venn diagram in Fig 1.9) The . x0 y0 w2 h1" alt="" AN INTRODUCTION TO MATRICES, SETS AND GROUPS FOR SCIENCE STUDENTS