Prove that in R with the usual topology, the only open sets connected by paths are the open intervals. Please be as clear as possible, showing the steps and definitions if necessary. Thank you a lot.
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- Use the set definitions A = {a, b} and B = {b, c} to express the elements in the powerset P(A) ∩ P(B). Use {} for the empty set Do not add any spaces in your answer. Example {f} not { f } {e,f,g} not {e, f, g} { , }Please help me solve this. I will be glad if you can explain in detail so I can understand. Thank you.I need help with this question It's about inequality notation.
- Suppose X and Y are subsets of some universal set. De Morgan's laws provide expressions for the complement of the union XUY and the complement of the intersection XnY. What about the complement of the set difference XY? You can answer this question by observing that X\ Y = Yen X. Which of the following expressions is equal to (X\Y) Yen X X ΠΥ XUY YUX X^ \ Y°Which of the following sets, given by the roster method, is equal to the set below given in set-builder notation? {x € Z | 0 < x² < 30} O A. {-30, -29, ..., -2,-1, 1, 2,..., 29, 30} O B. {1,2,..., 29, 30} O C. {-5, -4,-3, -2, -1, 1, 2, 3, 4, 5} O D. {0, 1, 2, 3, 4, 5}1. Give the domain of y=in set builder notation. 2. If a and b are real numbers such that aeach part below, indicate (by making suitable arguments) whether the following sets are open (yes/no), closed yes/no), and bounded (yes/no). Hint: You should probably graph the following sets. The straight line connecting the points (a, b) and (c,d) (including the endpoints) in R² where a < bPlease type the answer to the question and it should not be handwritten.Using the definition of an open set, prove that the interval (0, 1) is an open set.SEE MORE QUESTIONSRecommended 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,