Question. 1 and 2 pleas Problems1. Show that the real line is a metric space.2. Does d(x, y) = (x-y" define a metric on the set of all real numbers?3. Show that dix, y)= r-yl defines a metric on the set of all realnumbers.4. Find all metries on a set X consisting of two points. Consisting of onepoint.5. Let d be a metric on X. Determine all constants k such that (i) kd,(H) d+k is a metric on X.6. Show that d in 1.1-6 satisfies the triangle inequality.7. If A is the subspace of " consisting of all sequences of zeros and ones,what is the induced metric on A?8. Show that another metric d on the set X in 1.1-7 is defined by9. Show that d in 1.1-8 is a metric.1.2 Further Examples of Metric Spaces10. (Hamming distance) Let X be the set of all ordered triples of zerosand ones. Show that X consists of eight elements and a metric d on Xis detined by d(x, y) = number of places where x and y have differententries. (This space and similar spaces of n-tuples play a role inswitching and automata theory and coding d(x, y) is called the Ham-ming distance between x and y; ef. the paper by R. W. Hamming(1950) listed in Appendix 3.)11. Prove (1).12. (Triangle inequality) The triangle incquality has several useful conse-quences. For instance, using (1), show thatld(x, y)-d(z, w)s d(x, 2) + d(y, w).13. Using the triangle inequality, show that|d(x, 2)- d(y, z)s d(x, y),14. (Axioms of a metric) (M1) to (M4) could be replaced by other axioms(without changing the definition). For instance, show that (M3) and(M4) could be obtained from (M2) anddix, y)s d(z, x)+ d(z, y).15. Show that nonnegativity of a metric follows from (M2) to (M4),

Name: Question. 1 and 2 pleas Problems1. Show that the real line is a metric
Uploaded: 2024-03-20T07:07:50.000Z
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Description: Question. 1 and 2 pleas Problems1. Show that the real line is a metric space.2. Does d(x, y) = (x-y" define a metric on the set of all real numbers?3. Show that dix, y)= r-yl defines a metric on the set of all realnumbers.4. Find all metries on a set