Please fill the blanks of HW1 and HW2Thank you Defn: A metric space X is a collection of points with a distance function d such that(i) d(x,y) \ge 0 and d(x,y)=0 iff x=y.(ii) d(x,y)=d(y,x) for all r and y in X.(iii) d(x,y) \le d(x,z)+d(z,y) for all x,y,z in X.Here \le is the way to write less than or equal to in tex notation and \ge is the way to writegreater than or equal to.Iff means if and only if which is a two way implies symbol. So d(x,y)=0 implies x=y andalso x=y implies d(x,y)=0.HW1: Write the above definition of metric space using the quantifiers and impliessymbols. Prove the real line with the distance d(p,q)=|p-q| is a metric space using thequantifiers you learned in previous lessons. Hint: You need to prove all three rules of ametric space so each has its own part in your proof. Hint for (i): You may use |z| \ge 0 asa justification and |z|=0 implies z=0 as a justification. Hint for (ii): You may use Iz|=|-z| asa justification. Hint for (iii): You may use Ja+b| \le |a|+|b| as a justification taking a=x-zand b=z-y.HW2: Prove that a ball B(p,R) in the real line is an interval (p-R, p+R)Hint: First show B(p,R) is a subset of (p-R, p+R) using a sequence of steps as follows:1. Let x be in B(p,R) (1) given2. so d(p,x)

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Description: Please fill the blanks of HW1 and HW2Thank you Defn: A metric space X is a collection of points with a distance function d such that(i) d(x,y) \ge 0 and d(x,y)=0 iff x=y.(ii) d(x,y)=d(y,x) for all r and y in X.(iii) d(x,y) \le d(x,z)+d(z,y) for all x

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Answered: Please fill the blanks of HW1 and HW2…

Math

Advanced Math

Please fill the blanks of HW1 and HW2 Thank you

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.

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Please fill the blanks of HW1 and HW2

Thank you

Defn: A metric space X is a collection of points with a distance function d such that
(i) d(x,y) \ge 0 and d(x,y)=0 iff x=y.
(ii) d(x,y)=d(y,x) for all r and y in X.
(iii) d(x,y) \le d(x,z)+d(z,y) for all x,y,z in X.
Here \le is the way to write less than or equal to in tex notation and \ge is the way to write
greater than or equal to.
Iff means if and only if which is a two way implies symbol. So d(x,y)=0 implies x=y and
also x=y implies d(x,y)=0.
HW1: Write the above definition of metric space using the quantifiers and implies
symbols. Prove the real line with the distance d(p,q)=|p-q| is a metric space using the
quantifiers you learned in previous lessons. Hint: You need to prove all three rules of a
metric space so each has its own part in your proof. Hint for (i): You may use |z| \ge 0 as
a justification and |z|=0 implies z=0 as a justification. Hint for (ii): You may use Iz|=|-z| as
a justification. Hint for (iii): You may use Ja+b| \le |a|+|b| as a justification taking a=x-z
and b=z-y.
HW2: Prove that a ball B(p,R) in the real line is an interval (p-R, p+R)
Hint: First show B(p,R) is a subset of (p-R, p+R) using a sequence of steps as follows:
1. Let x be in B(p,R) (1) given
2. so d(p,x)<R
3. |p-x|<R
(2) defn of ball
(3) defn of distance on the real line
(4) |z|<R implies -R<z<R
(5)
(6).
4.
5.
6. x in (p-R,p+R)
Next show (p-R, p+R) is a subset of B(p,R)

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