Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H is convex.

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

Q: Let C be the line segment joining A(1,1,1) to O (0,0,0). A parameterization

A: Given that A(1,1,1) and O(0,0,0) Find a parameterization

Q: The set of all directed line segments that are equivalent to a given directed line segment PQ\is…

A: The set of all directed line segments that are equivalent to a given directed line segment PQ \ is a…

Q: Let u= and v= Show that is in Span (u, v} for all x and y.

A: Given u=4-1 , v=41 and b=xy. The vector b=xy is in span u,v if the system containing u,v and b is…

Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

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Q: 2- If ihe Velume is equal to Zevo, then the all yectors line in The Some plane. a.T b. F

A: Concept: Volume of an object is in 3 dimentional plane. If any of the dimentions is equal to zero,…

Q: If L is a line through 0 and if ŷ is the orthogonal projection of y onto L, then || ŷ|| gives the…

A: To state whether the given statement is true or false :  By definition, we know that, if L is a line…

Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…

Q: Prove the theorem 2.6.3 "if l is any line, then there exist at least one point P such that P does…

A:

Q: Prove that if A-C-D, B-F-C, and A-F-G (all in the same plane) and A, B, and C are noncollinear, then…

A: Given:  ACD , BFC and AFG all are in same plane and A,B and C are noncollinear , then G is the…

Q: Determine the dilation scale factor that would partition the directed line segment AB into the given…

A:

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Let u = and v = Show that is in Span (u, v} for all x and y. y -1 5

A:

Q: Let F = (z,2x, y) and C is the straight segment from (0,0,0, to (1,2,2). Then F• dř = %3D a) 7/2 b)…

A: Parametrize the line and then integration.

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: Write a proof using indirect proof in paragraph form. Given: Transversal t cuts lines a and b.…

A: Write a proof using indirect proof in paragraph form. Given: Transversal t cuts lines a and b. We…

Q: الموضوع show that if foret A frix by and ; YーZ is a continuous map then gFo relA gf,:X b Z

A:

Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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Q: The set [0, 1] with the discrete metric is compact. O True O False

A:

Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 =…

A: Given: r=x,y,z and r0=1,1,1 To describe the set of all points x,y,z such that r-r0=2 Solution:…

Q: Theorem 2.6.3. If l is any line, then there exists at least one point P such that P does not on l.

A: Axioms of incidence geometry: Incidence between points and planes: Three nonlinear points that do…

Q: The set [0,4] with the usual metric is sequentially compact. O True O False

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Q: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

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Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: A of a metric space (x, d.)

A:

Q: Prove or disprove that AS is congruent to SB (or that CS divides AB equally) if a line is drawn from…

A: Given, in △ABC, AB=AC=aBC=bBD=CDAD=h (altitude) To prove: CS does not divides AB in two equal parts.…

Q: Is the set S = {1,, 3, ...} with metric d(x, y) = |æ – y| for all x, y e S complete ? prove your…

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Q: IfL is a line through 0 and if ŷ is the orthogonal projection of y onto L, then || §|| gives the…

A: Given statement is

Q: Prove that if U and V are neighborhoods of x e R", then so are Un V and U U V.

A: A set A is said to be neighborhood of a point x∈ℝn , if there exist a positive real number r&gt;0,…

Q: graph'the Sorution to the follow imy inginequratityon animber line nur (x-2) (x+5) z O

A: Solving the problem by doing the graph.

Q: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

A:

Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

Q: Let Á = ry?a, + x³yzay, Evaluate A. dl along the path y = x2 from a(0,0,2) to b(3,9,2) Select one: O…

A: We will find out the required value.

Q: Let X = R, and M = {0}, then span (M) = %3D %3D OR O {0}

A: If, M = {0} then, span(M) = {0}

Q: Find all maps ¢:(Z,+) → (Q*, ×) satisfying O (a + b) = ¢(a) × ¢(b) Va, b e Z

A: We derive the function from the given properties.

Q: 7. Let ABCD be a parallelogram in the affine plane. Prove that the line connecting the midpoints of…

A: Given that, Let ABCD be a parallelogram in the affine plane. Prove that the line connecting the…

Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

A:

Q: Let C C R" be a cone. Show that C is convex if and only if C+C CC.

A: We prove the result by using the definition of convex sets.

Q: Use greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x…

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Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: 2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

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Q: Let X = {a, b, c} Define d: X2[0, 0) by d(a, a) = 0, d(b, b) = 0, d(c, c) = 0, d(a, b) = 1, d(a, c)…

A: Metric Space: Let X be a set and a function d :X×X→ℝ Then X, d is a metric space if d satisfies the…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Let D be the closed region in

A: Given,                    x2-2xy+y2+x+y=0   and      x+y+2=0