4. Let E' consist of all sequences in £2 satisfying: In other words, E' consists of only those sequences in E2 which never have two consecutive zeros. a. Show that o preserves and that E is a closed subset of E. inta of a are dense in '.

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: 1. Determine L{t*}. n! Ans. s*+1

A: We find the L{tn} using definition of Laplace Transform. We know that,

Q: 7. Let x E R and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

A:

Q: 9. Let a, b eR such that a a} I. (а, 5) A. I and II only В. П only C. I and III only III. (-00, b)…

A:

Q: 8. Let = {(0,²) In = {(0,²) n = 1,2,3,...}. Prove that n = 0.

A:

Q: Suppose that in AABC, mÃB + mĀČ < T. Prove that m < B +m < C < T.

A:

Q: 9. Let m and a be integers such that m > 1 and (a,m)= 1. Prove that if {r1, ...,r(m)} is a reduced…

A: Consider that  is a complete set of residues modulo m. It is given that gcd ( a,m) = 1The objective…

Q: 12. Suppose that k is a divisor of n. Prove that Z,/(k)~ Zp.

A:

Q: 6. Let a € Z and assume that gcd(a, 55) = 1. Prove that a40= 1 (mod 55). (Hint: Fermat's little…

A:

Q: 4. Let X be an infinite set, Define H:£= P(X) →R by O if A is finite u(A): if A is infinite. Then…

A:

Q: If S {1,...,n), for n ≥ 3, and A consists of all k-subsets of S, for n > k≥ 2, then (S, A) is a -…

A:

Q: 11. Prove the Cantor-Schroeder-Bernstein Theorem.

A: To Prove: The Cantor-Schroeder-Bernstein Theorem.

Q: 4. Let Y + 0 and let A be a collection of subsets of Y satisfying the following two properties: (a)…

A:

Q: Prove that if x is a real number such that x ≥ 0 and x ≤ € for all € E R where € > 0, then x = = 0.

A: This question is from real analysis.Answer given below.

Q: 7. Give an example of an indexed collection {An nE N} such that 1 An and U An = Aj = N.

A: Given that An|n∈ℕ To give an example of an indexed collection  An|n∈ℕ such that ∩n=1∞An=∅ and…

Q: 7. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

A:

Q: 44. Let A = {-2x+3 x € R} be a bounded subset of R. Find sup A. A B. - C. 1+ v2 D. 1- v2 E. None of…

A: First try is finding maximum  Luckily we got maximum therefore that became as an supreme

Q: 11.7 Let (rn) be an enumeration of the set Q of all rational numbers. Show there exists a…

A:

Q: Prove that the set B = {r € Q: 1² < 20 or r < 0o} is a cut.

A: In this question, we have to prove that the set,  B = x∈ℚ : x2 &lt; 2 or x &lt; 0 is a cut.

Q: Let S be the subset of (0, 1) consisting of numbers whose digits (in the decimal expansion)…

A: Given set is S=x∈0,1|digits in decimal expansion are alternate between odd and even We have to prove…

Q: A = {n € Z | (3k e Z)(n B = {n E Z | (3j e Z)(n t A = B.

A: Introduction: An even number is one that is divisible by two and has a remainder of zero. An odd…

Q: 6. Let E1, E2, E3, ... be subsets of R and suppose that f is defined and is uniformly continuous on…

A: Given f is uniformly continuous on E1,E2,E3,…. (a). Let n be any natural number. Given that f is…

Q: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

A:

Q: 4. Use mathematical induction to prove that 1+2+3+..+n =;n(n+ 1)

A:

Q: Let (E), be a countable disjoint collection of measurable sets. Prove that for any set A, 00 m* (AnU…

A:

Q: 3. Suppose (a,n) is Cauchy and that for all m, k 2 100, Jam - as 0. Does it follow that the…

A:

Q: . Is the power set P(H),

A: Given,   And the powerset of H: . * and *' on are defined by . We have to justify that the…

Q: lim inf A, U lim in ninf{A, U B} n inf{A, n B} %3D = lim inf A, n lim in + A, B, → B A, U B, → AUB

A: Given that to explain the statements which are not true as,

Q: Express the rate of inflation from 1985 to 2015 as a percent correct to two decimal places. Year…

A: To find - Express the rate of inflation from 1985 to 2015 as a percent correct to two decimal…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.