Exercise (4) 1. Show that: Suppose x is a linear space that is complete with respect to norms |.|. and , if there exists positive real numbers c such that x ≤ch, for all x € X, then||||| are equivalent and 2. Show that If (x) and (y) are bounded sequences in a normed space X and AER. Then ) and (x, y, are bounded sequences in X. در العراق
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- 6. Let A = {x|x ∈ R and x^2 − 4x + 3 < 0} and B = {x|x ∈ R and 0 < x < 6}. Prove that A ⊆ B.X vector space ||.|| and ||.||' if arbitrary norms are equivalent According to ||. || ' norm Cauchy sequence ||. || norm is also the Cauchy sequence. Research and ProveExercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x ‹ U there is € > 0 with (x − €, x + € ) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5,00); (3) (-∞0, ∞0); (4) { neN} U {0}; (5) The set Q of rationals. .
- Let > be a monomial order. If m_1 > m_2 > m_3 > ... is a decreasing sequence of monomials, prove that there exists a t such that m_t =m_{t+1} = ... (any decreasing sequence of monomials is eventually stationary)A set S is called 'denumerable' if there exists a bijection f : N → S. (a) Show that the set N>2 is denumerable because the function g: N → N>2, n > n + 1 is a bijection. (b) Prove that set Z>-3 = {-3, –2, –1,0, 1, 2,3, 4, 5, ...} is denumerable by building a bijec- tive function g: N → Z>-3•1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]
- The booking limits for four fare classes in a flight are given as b = (b₁,b2, b3, b4) = (15, 10, 5, 2). (a) What is the available capacity for these four classes, i.e., what is the booking limit for classes {1, 2, 3, 4}? [ Select] (b) From this available capacity, how many seats are protected for class 1, i.e., what is the protection level for class 1? 151. Let S = (0,7) U 27 +13... Define < on S by a1DRecommended 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,