Homework Answers
I have used all the 4
conditions of metric space and satisfied them. For d1 d3 and d4 i
have shown the example why they are not metric space. Hope this is
helpful for you. Thankyou
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11. For xe R' and y e R', define d,(x, y) = (x-y)2, Determine, for each...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The...
Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The Euclidean Metric of $2.2 is often called the 12 metric and written |x - y l2 for x,y e R. Show that the following three similar relations are also metrics: (a) the tancab, or 11 metric: lx-wi : = Σ Iri-Vil (b) the marirnum, or lo-metric. Ilx-ylloo:=max(zi-yil c) the comparison, or 10-metric. Ix_ylo rn (ri-Vi) where δ(t)- if t = 0
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Let B C R" be any set. Define C = {x € R" | d(x,y) <...
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Foundations of Analysis Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies δ(z,x)-0 and δ(x,y)-δ(y,x) for all z, y E X b...
Foundations of Analysis
Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies δ(z,x)-0 and δ(x,y)-δ(y,x) for all z, y E X but that is NOT a metric
Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies...
Show that for each x, y ∈ X, the metric e defined as e (x, y)...
Show that for each x, y ∈ X, the metric e defined as e
(x, y) = min {1, d (x, y)} is equivalent to the d metric.
1. Her x,y e X için e(x, y) = min{1. d(x,y)} şeklinde tanımlanan e met- riğinin d metriğine denk olduğunu gösteriniz.
Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we defin...
Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we define AY (2 E 9(X) : Y and Z have the same number of elements) (a) Prove that AY : Y є 9(X)} partitions 9(X). (b) Letdenote the equivalence relation on (X) that is associated with this partition (according to Theorem 11.4). If possible, find A, B, and C such that 1....
Define a relation R from R to R as follows: For all (x, y) E R...
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
Consider d on R defined by d(x, y) = ?|x − y|. (1) Show that (R,...
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
Let d: R XR + R be defined to be d(x, y) = |arctan(x) – arctan(y)]....
Let d: R XR + R be defined to be d(x, y) = |arctan(x) – arctan(y)]. Show that d is a metric on R.
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The Euclidean Metric of $2.2 is often called the 12 metric and written |x - y l2 for x,y e R. Show that the following three similar relations are also metrics: (a) the tancab, or 11 metric: lx-wi : = Σ Iri-Vil (b) the marirnum, or lo-metric. Ilx-ylloo:=max(zi-yil c) the comparison, or 10-metric. Ix_ylo rn (ri-Vi) where δ(t)- if t = 0
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Foundations of Analysis
Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies δ(z,x)-0 and δ(x,y)-δ(y,x) for all z, y E X but that is NOT a metric
Let X = {A, B,C,D}. ·Describe a function δ : X × X-> R that is a metric on X. . Describe a function δ : X x X-R that satisfies...
Show that for each x, y ∈ X, the metric e defined as e
(x, y) = min {1, d (x, y)} is equivalent to the d metric.
1. Her x,y e X için e(x, y) = min{1. d(x,y)} şeklinde tanımlanan e met- riğinin d metriğine denk olduğunu gösteriniz.
Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we define AY (2 E 9(X) : Y and Z have the same number of elements) (a) Prove that AY : Y є 9(X)} partitions 9(X). (b) Letdenote the equivalence relation on (X) that is associated with this partition (according to Theorem 11.4). If possible, find A, B, and C such that 1....
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
Let d: R XR + R be defined to be d(x, y) = |arctan(x) – arctan(y)]. Show that d is a metric on R.
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