Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B) ≠ ∅, prove that A∩∂ (B) ≠ ∅. Where ∂ (B) is the boundary of B.
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Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B) ≠ ∅, prove that A∩∂ (B) ≠ ∅. Where ∂ (B) is the boundary of B.
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- Find the distance from y to the plane in R³ spanned by the set (U₁, U₂} 6 - 2 4 y = -9 a. b. C. 43₁ 43, 43₁ d. 43, 1 195 2 195 3 195 3 391 -5 4₂ 1Prove that the norm induced by an inner product satisfies the parallelogram law. Using this property, prove that for p = 2, P is not an inner product space.Consider the 3D domain W = {(x, y, 2)| 2 2 > 0}. ;, y, z)| 2 2z > 0}. 2. Calculate x2 + y? + z? d(x, y, z) using / || dpdø d0 set up. W
- Consider the following subsets of the Euclidean plane R?: A = {(7.v) € R| - 3<33, 03ys8}; Az = {(r.v) e R* |y= r-}; A, = {(r. v) € R* | 5e* +7y" <11}. y = x -I + Tgl < 1}. (a) Which of the sets A1, A2, A3 CR? are open? (b) Which of the sets A1, A2, A, CR are closed? (c) Which of the sets A1, A2, Az C R? are compact? (Justify your answers concisely.)(b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у ||a) Determine if the vectors are linearly dependent or independent. b) Is it possible for the set (V1, v2,03, V4} to be considered as a base of R4 c) Consider the set H = Gen{V,02,03, vA) Is it a linear space? If so, how is its dimension found?a lineorly independent {x,, az} subset sa, xz,u} f Let be given US of a vecto r space. Show that the set the V space is lineorly indep en dest given a u ¢ So {dn, az} vector. pleose solue cleorly.2. Let V, W be subsets of R“ defined the following way: a а 2а + b a+b V = a,b ER W : a,b €R, a<0Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that d2 is a metric on ℝ2. b.) Draw the neighborhood N((0; 1) for d2, where 0 is the origin in ℝ2.Let V = R?, defined the addition by: (u1 tu2) + (V1 + v2)= ( U1 +v1 , uz tv2 + 1) With standard scalar multiplication. Is V a vector space justify ?. If V is an inner product space, then for any vectors a, ß in V and any scalar c (i) l|ca|| = |c||la||; (ii) |la|| > 0 for a + 0; (iii) (ælß)| < ||a|| ||ß|l; (iv) ||a + B|| < lla||+ I|B|I.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,