Mathematical Analysis I, 2004a

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12 Chapter 1. Set Theory (d) Consider f : N → N, withf(x) =2x for each x ∈ N. 3 By what was said above, f is well defined. It is one to one since x ≠ y implies 2x ≠2y. Here D f = N (the naturals), but D f ′ consists of even naturals only. Thus f is not onto N (it is onto a smaller set, the even naturals); f −1 maps the even naturals onto all of N. The domain and range of a relation may be quite arbitrary sets. In particular, we can consider functions f in which each element of the domain D f is itself an ordered pair (x, y) orn-tuple (x 1 ,x 2 , ..., x n ). Such mappings are called functions of two (respectively, n) variables. Toanyn-tuple (x 1 , ..., x n ) that belongs to D f , the function f assigns a unique function value, denoted by f(x 1 , ..., x n ). It is convenient to regard x 1 ,x 2 , ..., x n as certain variables; then the function value, too, becomes a variable depending on the x 1 , ..., x n . Often D f consists of all ordered n-tuples of elements taken from a set A, i.e., D f = A n (cross-product of n sets, each equal to A). The range may be an arbitrary set B; sof : A n → B. Similarly, f : A × B → C is a function of two variables, with D f = A × B, D f ′ ⊆ C. Functions of two variables are also called (binary) operations. For example, addition of natural numbers may be treated as a map f : N × N → N, with f(x, y) =x + y. Definition 4. ArelationR is said to be (i) reflexive iff we have xRx for each x ∈ D R ; (ii) symmetric iff xRy always implies yRx; (iii) transitive iff xRy combined with yRz always implies xRz. R is called an equivalence relation on a set A iff A = D R and R has all the three properties (i), (ii), and (iii). For example, such is the equality relation on A (also called the identity map on A) denoted I A = {(x, y) | x ∈ A, x = y}. Equivalence relations are often denoted by special symbols resembling equality, such as ≡, ≈, ∼, etc. The formula xRy, whereR is such a symbol, is read “x is equivalent (or R-equivalent) to y,” 3 This is often abbreviated by saying “consider the function f(x) =2x on N.” However, one should remember that f(x) isactuallynotthefunction f (a set of ordered pairs) but only a single element of the range of f. A better expression is “f is the map x → 2x on N” or “f carries x into 2x (x ∈ N).”
§§4–7. Relations. Mappings 13 and R[x] ={y | xRy} (i.e., the R-image of x) is called the R-equivalence class (briefly R-class) ofx in A; it consists of all elements that are R-equivalent to x and hence to each other (for xRy and xRz imply first yRx, by symmetry, and hence yRz, by transitivity). Each such element is called a representative of the given R-class, or its generator. Weoftenwrite[x] forR[x]. Examples. (a ′ ) The inequality relation < between real numbers is transitive since x
Page 1: The Zakon Series on Mathematical An
Page 4 and 5: Copyright Notice Mathematical Analy
Page 6 and 7: vi Contents 7. Intervals in E n ...
Page 9 and 10: Preface This text is an outgrowth o
Page 11: About the Author Elias Zakon was bo
Page 14 and 15: 2 Chapter 1. Set Theory For any set
Page 16 and 17: 4 Chapter 1. Set Theory The symbols
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Page 22 and 23: 10 Chapter 1. Set Theory Then R[1]
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Page 36 and 37: 24 Chapter 2. Real Numbers. Fields
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Chapter 3 Vector Spaces. Metric Spa
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§§1-3. The Euclidean n-Space, E n
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§§4-6. Lines and Planes in E n 73
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§7. Intervals in E n 77 2. In part
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§7. Intervals in E n 79 Corollary
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§8. Complex Numbers 81 We make som
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§8. Complex Numbers 83 where x and
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§8. Complex Numbers 85 Use it to f
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∗ §9. Vector Spaces. The Space C
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∗ §9. Vector Spaces. The Space C
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∗ §10. Normed Linear Spaces 91 E
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∗ §10. Normed Linear Spaces 93 P
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§11. Metric Spaces 95 §11. Metric
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§11. Metric Spaces 97 For a fixed
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§11. Metric Spaces 101 Reverse all
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§12. Open and Closed Sets. Neighbo
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§13. Bounded Sets. Diameters 109 D
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§13. Bounded Sets. Diameters 111 s
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§13. Bounded Sets. Diameters 113 4
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§14. Cluster Points. Convergent Se
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§17. Cauchy Sequences. Completenes
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150 Chapter 4. Function Limits and
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252 Chapter 5. Differentiation and
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Index Abel’s convergence test, 24
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Index 343 arcwise-, 211 curves as,
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Index 345 as a normed linear space,
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Index 347 bounded, 111 continuity o
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Index 349 limits in one variable, 1
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Index 351 Ordered pair, 1 inverse,
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Index 353 nondecreasing sequences o
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Index 355 Euclidean spaces, 87 norm
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Mathematical Analysis I, 2004a

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