Appendix A Parentheses, Their Proper Pairing, and Associativity A.1 Proper Pairing of Parentheses We all know how to deal with parentheses in mathematical expressions; what is their role, how do they pair, what are the rules of pairing, and so on. Let us try to formalize all this. To begin with, we have an alphabet of two symbols, the left parenthesis, written “(” and called here in short “lp”, and the right parenthesis “)” called “rp”. Then we have finite strings of such parentheses. Not all such strings are, however, of interest to us. We want to look at only those that consist of parentheses that are ‘properly’ paired, or matched. Intuitively, we know what proper matching means: for every lp there is precisely one matching rp and vice versa, every lp is to the left of its matching rp, and no two matching pairs interlace. How do we say all this in the language of algebra? Well, one way is to start with an alphabet A = {(,)}, consisting of the set of the two symbols denoting the two parentheses, and the set of all finite sequences (or strings) of these symbols.1 Let pi denote the entry in the ith position of a string (counted from the left); it may be either an lp or an rp. A string is of length n if it has n such entries p1 , p2 , . . . pn . It is convenient here to treat a string s as an indexed set S, with the entries pi as its members: S = {pi }. Let f : S → S be a partial function on S that associates an lp of S with an rp of S.2 Then, to say that f accomplishes the matching we have in mind is to say that it meets the following four conditions: (i) f is one-to-one (ii) if pj in S is an rp then there is an lp, say pi , such that f (pi ) = pj (iii) if f (pi ) = pj then i < j (i.e., pi is 1

Here, and in the subsequent discussions on parentheses, I follow Manaster [3, pp. 5–6]. A partial function from a set A to a set B is a function from a subset of A to a subset of B.We talk of a partial function here because it is defined over only some of the members of B, the right parentheses. 2

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152

to the left of pj in s), and (iv) there is no interlacing of lp–rp pairs, i.e., if pi and pj are lps with i < j, and f (pi ) = pi , f (pj ) = pj  , and if j < i , then j < j  < i .3 If such a function exists for a string then we say that the string has a proper pairing. It is such a proper pairing of parentheses that we rely on to specify the order in which operations have to be performed in an algebraic expression. Crucial in this respect is the fact that there can not be two alternative proper pairings for the same string of parentheses.4 Theorem A.1.1 A finite string of left and right parentheses admits of at most one proper pairing. P ROOF : To check that it is indeed so, we argue by induction on lengths of strings. We do this by first identifying a pair formed by an lp followed immediately by an rp, suppressing the pair, and then relating the resulting string of smaller length to the original one. Let pi be the leftmost rp in a string. Assuming that there is a proper pairing of the parentheses, there is then an lp, say pi , i < i, for which f (pi ) = pi .5 Then it must be that i = i − 1. For, suppose there is j  , i < j  < i, such that pj is an lp, pairing with pj , an rp (necessarily) to the right of pi (i.e., j > i). This will amount to an interlacing, contradicting our assumption that we have a proper pairing here. Thus i = i − 1. In all, if the original string has a proper pairing then there is in it an lp–rp pair, pi−1 pi . Now the inductive part. Proper pairing requires an even number of parentheses in a string. So, we need to look at strings of lengths 2(n + 1) for n = 0, 1, 2 . . .. For n = 0, there is just one string with proper pairing—the string “()”, and this pairing is unique. Let us now assume that for any n, i.e., for any string of length 2(n + 1), if a proper pairing exists then it is unique. Moving from n to n + 1, consider now a string p1 p2 p3 . . . p2n+2 p2n+3 p2n+4 of length 2((n+1)+1) = 2n+4 that has a proper pairing. As argued earlier, this string has an innermost pair consisting of pi−1 , an lp, paired with pi , the left most rp of the string. Suppressing this pair, we get a string for which there is at most one proper paring by hypothesis. On the whole, the original string then also has at most one proper pairing. It then follows by induction that for any finite string of parentheses, there is at most one proper pairing. This would be a good point to look at a text such as Manaster [3, Chapter 1], whose line of argument I have used here, for additional material related to logic and formal languages. 3 ( (

( ( )

)

For example, the pairings resulting in pi pj pi pj  are not allowed, whereas those resulting in )

)

pi pj pj  pi are allowed. 4 In organizing a mathematical expression in parentheses, we have in mind a certain order in which the operations are to be performed, and the parentheses are meant to indicate that order. But if another proper pairing were possible, then some one evaluating that expression could go by this other pairing and get a result altogether different from the one intended. 5 It can not be an rp since we have already picked up the leftmost.

A.2 Parentheses and the Associative Law

153

A.2 Parentheses and the Associative Law Consider a binary operation given on a set S. Adopting the multiplicative notation for the operation, the associative law says that for any three elements x1 , x2 , x3 ∈ S, their products formed in the two possible ways for the given order are equal: (x1 x2 )x3 = x1 (x2 x3 ). So, if the given operation is associative, we omit the parentheses and simply write x1 x2 x3 for the unique product. Now, if we have more than three elements, we can form their product in successive steps in various different ways. Thus for four elements, x1 , x2 , x3 , x4 , for instance, we get (x1 x2 )(x3 x4 ), (x1 (x2 x3 ))x4 , ((x1 x2 )x3 )x4 , amongst several others. Clearly, the second and the third ones are equal—applying the associative law to the first factor of the former, we get the latter. What about the first one? Well, again by the associative law, we have (x1 x2 )(x3 x4 ) = ((x1 x2 )x3 )x4 . We wish to show that the associative law implies equality of such products for any finite number of elements. To be more specific, for elements x1 , x2 , . . . , xn ∈ S, n ≥ 3, consider their products formed in various different ways, each of these in several successive steps indicated by parentheses. We use induction to show the following. Proposition A.2.2 For the given order in which the elements x1 , x2 , . . . , xn appear, their product under an associative binary operation is in the end the same for any n ≥ 3, irrespective of the differences in the successive steps. This unique product we simply write as x1 x2 . . . xn , leaving out the parentheses. Let us say that the proposition is true for elements numbering less than n. Now, for n elements, any particular way of forming their product will in the last step have two factors having less than n elements in each. The product thus takes the form (x1 x2 . . . xk )(xk+1 . . . xn ) for some k < n. Likewise, some other way of forming the product yields (x1 x2 . . . xl )(xl+1 . . . xn ) for some l < n. Without any loss of generality, let us say that l < k. Then, by the associative law, we have (x1 x2 . . . xk )(xk+1 . . . xn ) = (x1 x2 . . . xl )(xl+1 . . . xk )(xk+1 . . . xn ) = (x1 x2 . . . xl )(xl+1 . . . xn ) . In other words, any two different ways of forming the product give the same result. Given that the proposition is true for n = 3 (the associative law), it follows that it is true for all n ≥ 3.1 Aside 4 There are some fine points that we need to take notice of here. In many situations, associativity is not directly stipulated for the given binary operation. It inherits it from that of another operation in terms of which it is defined. Consider for instance the case of union of sets. We say that for two sets S1 and S2 , their 1 For more on this and other related issues, see Kochend¨orffer [2, pp. 1–4] and Artin [1, pp. 40–41].

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union S1 ∪ S2 = {x|x ∈ S1 or x ∈ S2 }. So defined in terms of the set membership relation ‘∈’ and the logical connective ‘or’ (a binary operation on propositions), the union operation is associative because the connective ‘or’ is so by definition. We may thus invoke Proposition A.2.2 for the union operation applied to n sets and write the union as S1 ∪ S2 ∪ · · · ∪ Sn . We may alternatively define union of n sets in general as S1 ∪ S2 ∪ · · · ∪ Sn = {x|x ∈ S1 or x ∈ S2 or · · · or x ∈ Sn }. In this definition, Proposition A.2.2 is invoked at the inner logical level for the connective ‘or’, and S1 ∪ S2 becomes a special case. Of course, whether we define union this way or we first define it as a binary operation and then through repeated applications for n sets, effectively the same purpose is served. Finally, there is also a point about notation. In S1 ∪ S2 , the symbol ‘∪’ denotes a binary operation whereas in S1 ∪ S2 ∪ · · · ∪ Sn , its repeated presence denotes an n–ary operation. Considering the close connection between the two and the resulting convenience, this ambiguity is harmless. ♥

References 1. Michael Artin. Algebra. Prentice–Hall, New York, 1991. 2. Rudolf Kochend¨orffer. Group Theory. McGraw-Hill, New York, 1970. 3. Alfred B. Manaster. Completeness, Compactness, and Undecidability. Prentice–Hall (India), New Delhi, 1975.

Index 2–port passive resistive network, 86 3–port, 88 absorptive, 62 additivity, 12 algebra, 12 analog filter, 145 antisymmetric, 61 Archimedean Principle, 69 Archimedes, 69 associative, 49, 62 associative law, 49 associativity, 49 automorphism, 52, 85, 108 axiomatics, 4 basis vector symmetry-adapted, 134 block diagonal form, 118 block–diagonalizable, 98 simultaneously, 98 block–diagonalize, 97 Boolean algebra, 64 cancellation laws, 50 cardinality, 95 Cartesian product, 46 cascade, 11 causal, 16 causality, 16 Cayley table, 47 character, 122 characters, 122 circuit active, 44 passive, 44 closure, 48

codomain, 46 commutative, 49, 62 complemented lattice, 63 compression, 76 concatenation, 71 conceptual metaphor, 77 conjugacy classes, 52 conjugate, 52 convolution, 9 dyadic, 25 convolutional algebra, 21 coordinates, 54 cyclic group, 29 cyclic interchanges, 88 cyclic matrix, 30 dc excitations, 86 decoding, 76 Dedekind, 75 definition, 43 nature of, 44 delays, 72 diagonalize, 97 differentiation, 73 direct sum, 58 direct sum decomposition, 58 discrete cosine transform(DCT), 148 discrete finite transform (DFT), 147 Discrete Fourier Transform(DFT), 24 discrete signal transforms, 148 distributive lattice, 64 domain, 46 source, 77 target, 77 157

Index

158 element complement, 63 greatest, 62 least, 62 smallest, 62 largest, 62 empirical, 1 encoding, 76 equilateral triangle, 83 equivalence class, 48 equivalence relation, 47 Erlangen Programme, 53 fast algorithms, 147 Fast Fourier Transform (FFT), 147 filter design, 145 finite impulse response(FIR), 20 formal, 1 Fourier transform, 24 general linear group, 108 geometric symmetry, 82 greatest lower bound, 62 group, 50 abelian, 50 finite, 50 infinite, 50 group theoretic techniques, 82 group theory, 81 groups and their representations, 82 Hamilton, 75 harmonics, 144 homogeneity, 12 homomorphism, 52 hypersets, 77 idempotence, 62 idempotent, 60 identity, 49 identity element, 49 impulse, 8 shifted, 8 index set, 8

indexing, 45 infimum, 62 infinite impulse response(IIR), 20 infinitesimals, 77 inner product, 33 input, 8 integral domain, 21 invariant subspace, 112 invariant under G, 112 inverse, 49 irreducible characters, 122 irreducible subspace, 116 irreps, 126 isometry, 53 isomorphism, 52 isosceles triangle, 85 kernel, 120 Langrange’s Theorem, 52 Laplace transform, 24 lattice, 62 complemented, 63 distributive, 64 non-distributive, 64 least upper bound, 62 linear, 12 linear transformation projection, 115 linear transformations symmetry of, 93 locations, 74 lower bound, 62 greatest, 62 many-to-one, 8 map, 46 matrices of the representation, 108 matrix representation, 106 of groups, 105 measurement, 69, 70 representational view of, 70 metaphor conceptual, 77

Index metaphors, 69, 76 mixed radix number system, 95 model, 74 modeling, 69, 72 moments, 74 morphisms, 52 multiplication table, 47 mutually orthogonal, 34 nature of definition, 44 network theory, 83 non-distributive lattice, 64 non-recursive, 19 nonstandard analysis, 77 one–to–one, 46 one-to-one, 8 onto, 46 operation arity of, 47 induced, 49 n-ary, 47 orthogonality, 124 Orthogonality Theorem, 119, 122 oscilloscopes, 73 output, 8 partial order, 61 partially ordered set, 61 partition, 48 partners, 134 Peano postulates, 75 perfect copies, 71 periodic signal, 82 permutation matrices, 87 permutations, 84 poset, 61 projection, 59 realized, 19 recursive, 19 reflexivity, 48 relation, 46 binary, 46

159 representation, 2, 71 complete reduction of, 130 completely reduced form, 117 equivalent, 111 faithful, 108 irreducible, 113 reducible, 113 regular, 110, 125 restriction of, 113 trivial, 110 representation of a finite group G, 108 representation space, 108 representational measurement theory, 72 resistive 2–port, 82 scalar zero, 8 scalar product, 124 Schur’s lemma, 119 sequence scalar, 8 set complement, 57 empty, 45 index, 45 nonempty, 45 power, 45 shift operator, 8 shift–invariant, 15 shrink, 112 signal, 1 space, 3, 5 complement, 58 direct sum decomposition of, 58 signal, 7 spectral components, 144 state equations, 146 Stone’s Representation Theorem, 64 structure, 6 empirical, 71 formal, 71 mother, 6 structure-preserving, 85

Index

160 structure-preserving mapping, 13, 14 structures algebraic, 6 isomorphic, 26 relational, 6 topological, 6 subgroup, 51 subset, 44 proper, 45 subspace direct sum, 58 sum, 57 supremum, 62 symmetry, 48, 53, 81 geometric, 82 symmetry-adapted basis vectors, 134 system, 7 convolutional, 9 systems formal, 76 natural, 76

time–invariant, 15 tortoise, 69 transducer, 72 transform domain, 31 transformation idempotent, 60 transforms, 32 transitivity, 48 translate, 94 translation operator, 94 translation–invariant, 15

temporal events, 82 time branching in, 75 flow of, 75

Walsh–Hadamard transform, 25 waveguide, 83

unit step, 8 unitary matrices, 112 upper bound, 62 least, 62 veil of familiarity, 75 veil of ignorance, 75

Z–transform, 20