1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that is a σ‐algebra and contains the two half‐open intervals A and B.
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1. Let Ω = (0, ∞). Let A = (3, 8] and B = (5, 10]. Find the smallest collection of subsets of Ω that is a
σ‐algebra and contains the two half‐open intervals A and B.
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- Which of the following operations on the set Z has more than one left neutral element and no right neutral element? (a) a ° b = b - a (b) a°b = b (c) a°b = a + ab - 1 (d) a°b = ab (e) Such a prescription does not exist.Let X be a set with an operation *: X² → X such that (i) x * x = x for all x E X, (ii) x * (y * z) = z * (x * y) for all x, y, z E X. Show that * is associative and commutative.Let p ≥ 1 and lp be the set of all sequences x = (x1, x2, · · ·) of real numbers suchthatkxkp =Xi|xi|p1/p< ∞.Show that lp with p 6= 2 is not an inner product space.
- 2) Let n E N. Suppose An = {n, n+ 1, n +2, ...} and T = {0, An}neN• (a) Show that t is a topology on N, (b) What are closed sets in t? (c) List the open sets containing the positive integer 6. (d) What is the closure of {4} and {2,4,6, 8, . ..}?(2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. (a) Show that the only open sets that are both closed and open in T are C and 0Which of the following operations on the set Z has more than one left neutral element and at least one right neutral element? (a) a°b = b - a (b) a°b = b (c) a°b = a + ab - 1 (d) a ° b = ab (e) Such prescription does not exist.
- 1. Find the union C1 U C2 and the intersection C1n C2 of the two sets C1 and C2, where (a) C1 = {0, 1, 2, }, C2 = {2,3, 4}. (b) C1= {x:0 < x <2}, C2= {x :12. Let u₁ -8 ] 10 4 Determine if x is in the span of ₁, ₂, and 3. If x is in the span then express it as a linear combination of ₁, ₂, and ū3. ● 5 = ) U₂ = ūz = 3 -11] and x =Q1: (a) If (M, d) be a metric space and AC M. Prove that A is the largest open set in M that contained in A. then give an example to show that (A U B) # A* U B. (b) let (M, d)be a metric space and let A ≤ M. Then (AB) ≤ĀNBRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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