(a) Suppose that A and B are nonempty sets of real numbers which arebounded above, and suppose that sup A < sup B. Prove that thereexists an element b E B so that b is an upper bound for A.(b) Give examples of nonempty sets A and B of real numbers which arebounded above so that sup A ≤ sup B and so that no element of Bis an upper bound for A.

(a) Suppose that A and B are nonempty sets of real numbers which are bounded above, and suppose that…

Answered: (a) Suppose that A and B are nonempty…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an interior point of A, (b) x is a boundary point of A or (c) 푥 is an exterior point of A.
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Does there exist a 1-1 and onto function that maps the integers to the rationales? Choose the best response below. Yes, but the function is not easily written in a "normal" way. Yes, and thus you can easily write it explicitly. No, because you can't write it explictly. No because the two set don't have the same cardinality.
Without proving anything, determine if the following statements are true or false. For any false statements, give an counterexample. (a) A finite, nonempty set of real numbers always contains its supremum. (b) If a < L for every element a in the set A, then sup A < L. (c) If A and B are sets with the property that a < b for every a E A and b e B, then sup A < inf B. (d) If sup A = s and sup B = t, then sup(A+ B) = s+t. Here and elsewhere, the set A+B is defined as A+B = {a+b:a € A, and b e B}. (e) If sup A < sup B then there is an element of B that is an upper bound for A.
a) It is known that Q ͠ N (Rational Numbers have the same power as the Natural Numbers set N.). So show that QxQ ͠ N. ( QxQ have the same power as the N ) (Definition: Let A and B be two sets.If there is a one-to-one function from A to B and at least one overlying function, it is said that A set has the same power as set B. and shown to be A ͠ B) (It's abstract mathematics, please can you write step-by-step solutions I have a shortcoming in this regard.)
A number r ∈ R is called algebraic provided for some n ∈ N there exists numbers a0, a1, … , an ∈ Z such that r satisfies p( r ) = 0 where p ( x ) = a0 + a1x + a2x2 + ⋯ + anxn. A number r ∈ R is called transcendental provided it is NOT algebraic prove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.
All
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