1. (a) Let A be the subset of R defined by A := {1-r:neN}. Find all cluster points of A and justify your answer. (b) Let S cR and let e e R be a cluster point of S. Prove that for each e > 0 the set W := sn(c-E,e +e) has infinitely many elements.
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- 4. Let (X, be a partial ordered set. Show that a subset A of X has at most one least upper bound.2.28 let xa be a translation of R Then (1) La is a continuous bijection from onto R (ii) The image of an open set under Da is an oper set. of (iii) let & be an open set. The component interval +a are exactly The images of the component intervals of the set & under translation a the goal of this section is to establish the following result.Let A and B be subsets of R which are bounded above. a. Show that sup(AUB) = max {sup(A), sup(B)}. Let y = sup (A U B), a = sup (A), and ß = sup (B)
- what is the rank of I-XX²) Xnxp of rank k. where Inxn1. (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)]After 25 years, your savings account has now grown to $16,200 and has an interest of 6.1% that compounds continuously. How much did you deposit at the beginning of the account
- How large should n be to guarantee that the Trapezoidal Rule approximation to x - 12x- 48x² + 2x + 5) dæ is accurate to within 0.1. n = How large should n be to guarantee that the Simpsons Rule approximation to 2 1(-a - 12a - 48z? + 2x + 5) dx is accurate to within 0.1. n = Hint: Remember your answers should be a whole numbers, and Simpson's Rule requires even values for n Question Help: Video 1 D Video 2 Submit Question P Type here to search hpLet n be a positive integer and let Sn be any set with |Sn|= n. Define Dn to be the digraph with V (Dn)= P (Sn) , the set of all subsets of Sn where (X, Y) element of A(Dn) if and only if X contains Y properly as a subset. a) Make a pictorial representation of D3 b) Prove that D has a unique source. c) Prove that Dn has a unique sink. d) Find a necessary and sufficient condition for Dn to have carrier vertices. e) Find a formula for the size Dn in terms of n. f) Prove that D has no circuit.6. if x C y. (a) (b) Let R = {a: a is a cut}.Let x, y, R. We say that a3 Consider [0, 1] × [0, 1] with dictionary ordering. Find Ã, when A = {(x, y) : 01. Check that the following functions on R² are norms: 1/p a) | (1,72) ||,= (1z1P" + \#a!P), 1spRecommended 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,