Justify rigorously the given set is open in R³. (a) A = {(x, y, z) |x>0, y<0, x² + y² +2²> 9}. (b) B = {(x, y, z) | x² + y² < 25,0
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Q: tify rigorously the given set is closed in R³. A = {(x, y, z) | x ≥ 0, y ≥ 0, z ≥ 0, x+y+z ≤ 12}. B…
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Q: Justify rigorously the given set is closed in R³. (a) A = {(x, y, z) | x ≥ 0, y ≥ 0, z≥0, x+y+z≤…
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- For each of the sets find the set of boundary points and the set of interior points. Also state if the set is open (explain), closed (explain), bounded, and compact (explain). (a) [0, 1] U {4} (b) {(x, y) = R² : ² + y² = 1}Q1) Choose the correct answer 1- Let X be a non-empty set and r = (-0,a)| a € R) URU(4) then r is called: (11) Indiscrete (1) Discrete (I) Left ray (IV) Right ray 2- Let X be a topological space and AS X, then the largest open set contained in A is: (1) Ext (A). (1) A". (IV) Ā (1) A'. 3- Two subsets A and B of the space X are called pair of disconnection of X if (1) AnB = 6, AUB = x () AnB + (m) AUB = X 4- Let X = (a, b, c), and 1= (X, , [a), (b,e}}, then the subset {c) is : (1) open and closed. (IV) All. (1) open. (I) closed. (IV) not open not closed Q2) True or false (1) B = {[a), (b), (c} is a base of the discrete topology on X = (a, b,c). (2) r= {x,. (x,y). (x,z). [y.z)} is not a topology on X = {x, y,z}. (3) For X = (a, b, c), the excluding point topology is E, = {X, g. (a), (c). (a, c), (a. b}). (4) For a discrete topology on X and o + ACX, the cdosure of A is A. (5) For an indiscrete topology on X = {a, b, c) and A = {a, b), the limit points of A is (a, b). (6) For the usual…2. A robot performs the following operations on a set of points (x,y) in the 2D plane to check the movement of objects where (x,y) E Zxz (a, b)A(c,d) = (c + a. 2db) Algebraic characteristics of the Summarize and Find the Abstract operation above.
- (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A)-(1)-(). and v= Describe the set Span {u, v, 3u, 2u - 5v, 0} geometrically, where u = 04. Show that R¹ and R² are not homeomorphic. How about R¹ and R" for n ≥ 2?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,