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Midterm1Draft5FormOmegaAnswerKey

Midterm1Draft5FormOmegaAnswerKey - FORM Ω FORM Ω FORM Ω...

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Unformatted text preview: FORM Ω FORM Ω FORM Ω FORM Ω FORM Ω FORM Ω 1 Please answer all questions. Each of the four questions marked with a big number counts equally. Designate your answers clearly. Recall the following definitions, concerning subsets of R N : • a set is ‘closed’ if it contains all of its cluster points (limit points). • a set is ‘open’ if, for each point in the set, there is an N-dimensional epsilon1-ball, epsilon1 > 0, (neighborhood) centered at the point, contained in the set. • a set is ‘bounded’ if it can be contained in a cube of finite size, centered at the origin. • a set is ‘compact’ if it is both closed and bounded. • a set is ‘convex’ if for every two points in the set, the set includes the line segment connecting them. 1 1. Is the following subset of R 2 closed? open? bounded? compact? convex? Explain your answer. L = 30 o line through (0, 2) = { ( x, y ) | ( x, y ) ∈ R 2 , y = 1 2 x + 2 } . Suggested Answer: Closed: contains its limit points; Not open: contains no open balls in R 2 ; unbounded: a line goes on forever; Not compact: closed but unbounded; Convex: contains the line segment between any two points of the line....
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Midterm1Draft5FormOmegaAnswerKey - FORM Ω FORM Ω FORM Ω...

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