R.1. Use the Schröder-Bernstein Theorem to prove that the intervals [0, 1) and (1, ∞) on the real number line have the same cardinality.
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Q: Problem 3. Let a be an integer that is coprime with 240. Prove that 240|a4 – 1. -
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- One of the super-powers of linear algebra is that it can be applied everywhere. In the next question, you will use linear algebra to solve a non-linear problem. Given a list of n points (x,y) R with distinct a, values, it is sometimes necessary to find a polynomial curve p(x) Pn 1(R) passing through all the points. One method to do so, uses linear algebra as follows. (We will learn another method, called Lagrange interpolation, later in the course.) Consider an arbitrary polynomial p(x) = ao + a₁x + a₂x²+...+ an-1" We consider the coefficients as unknown. Each point (U) gives a linear relation among the coefficients. The linear system is given: ao + a₁x₁ + a₂x² + ... + an_1x₁ ao + a₁x2 + a₂x² + n-1 + an 1x2 1 ao+an + ax² + + an 1 xn = Yı Y2 Yn 1. Use this method to find a quadratic polynomial passing through the points (0, 3), (1, 2), (2, 3). 2. Use this method to find a cubic passing through through the points (0,5), (1,9), (2, 11), (3,5). 3. Consider the example of the three points…Problem 3. Let a be an integer that is coprime with 240. Prove that 240|a4 – 1.Problem 1. A set I ≤ R is called an interval if for all triples of real numbers x, y, and z such that x < z < y, if x € I and y € I then z € I. Prove that every interval takes exactly one of the following nine forms: (-∞,b), (-∞,b], (a,b), (a,b], [a,b), [a,b], (a, ∞), [a, ∞), or (-∞, ∞). (Hint: You have to consider an arbitrary set I satisfying the condition in the problem and prove something about it. Consider cases according to whether I is (unbounded above / bounded above and contains its sup / bounded above and does not contain its sup), and similarly for below. You should have 3-3 = 9 cases.")
- give a complete statement of the definition ... then show how g(n) = O(h(n)) follows from this definition give rigourous proof derived directly from definition of O notationProblem 3. Show that if x is a real number and n is an integer, then 3(a) x15. The following puzzle appeared without attribution in the Spring 1989 Newsletter of the Northeastern Section of the Mathematical Association of America. What is the flaw in this argument? Euler's identity? ei = (ei)2π/2 = (e²ni) 0/2π = (1) 0/2π = 1.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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