• {1/n :n € N} • {2 – 1/n : n € N}. • {e";r € Q}. • {r² : 0 < ¤ < 2}. n(1 – 1/n, 2 + 1/n) where n E N U[1/n, n] where n E N.

Q: 3. List all of the permutations of S4. Find each of the following sets: a) {0 € S4:0 (1) = 3} b) {0…

A: From the above details we have,

Q: 6. List every generator for the subgroup of order 8 in Z32.

A:

Q: 2. Consider the following labeling scheme for the set of rational numbers Q = {†, where a E I, and…

A:

Q: ii) Prove that if f is continuous on [a, b], ƒ(x) > 0 on [a, b], and if f(c) > 0 for some ce (a, b),…

A:

Q: 2. Prove that a 2 x 2 matrix A with entries in Z26 is invertible if and only if gcd(det(A), 26) = 1.

A:

Q: dy 6. + 2xy² = 0 dx

A: dy/dx + 2xy2 = 0 here we have to find the solution of this euqaiton  further step given below this…

Q: iii Let f(x) be any continuous function and f(c) > 0 for some c. Find an interval (a, b) containing…

A: Please check the answer in next step

Q: 1. Find the sup and inf of the following sets. {1/n :n e N} • {2 – 1/n :n € N}. • {e";r € Q}.

A: Supremum: Greatest member of a set, if it exist, is the supremum of the set. Infimum: Smallest…

Q: 2. Give an example of an unbounded sequence {an} of negative terms such that an does not diverge to…

A:

Q: Show {5, 10, 15, 20, - - ·} is countable. (Hint: Use the function f(n) and onto). = 5n and show it…

A: 4 Given set is 5,10,15,20,....... We have to show that the set 5,10,15,20,...... is countable. Let a…

Q: Let S = R\{-1} and define a binary operation on S by a*b = a + b + ab. Prove that (S, *) is an…

A: 2) S=R∖-1 binary operation defined by a*b=a+b+ab

Q: 3. Prove that if a E Z then the only positive divisor of both a and a + 1 is 1.

A:

Q: #1. ii. Thanks

A:

Q: • Write 2.4020202 -.. as a fraction of two integers. • Can the number 3.73747879710711 -.. be…

A:

Q: 4. Let a1 = v2 and let an for n > 2 be defined recursively by the formula = Ip V2+ van. an+1 = Prove…

A:

Q: Find the order of each of the following elements. [from #2, 4.5] a) 5 € Z12 b) -i E C*

A:

Q: c an sin d an = n (1 +(-1)")

A: As given by the question There are given that the sequences: an=sinnπ4 Now, From the properties of…

Q: iii) Prove that if f is continuous, f(x) > 0 on [a, b], and f(x)dx = 0, then f is identically 0 on…

A:

Q: 3. Use the MVT to show that the function f(x) = 1/x³ is uniformly continuous on the interval 3,00).…

A:

Q: Use the MVT to show • Prove that the derivative of f(x) = k, on [a, b], where k is constant, is…

A: We know that the mean value theorem states that: If f is a continuous function on a,b, f is…

Q: 5. Prove that if {a,}f° converges to 1, then {an}° converges to 1. Hint: You may use a logarithm…

A: If an1∞ converges to 1, then an1/21∞ converges to 1.

Q: 3. Show that the sequence an = 1+ is a Cauchy sequence. Hint: Every convergent sequence is a Cauchy…

A: Cauchy sequence and it's convergence

Q: 8. Prove that Zp has no nontrivial subgroups if p is prime. [#26, 4.5]

A: Follow the steps.

Q: i Let f(x) = r° and c= 0.01. Find an interval (a, b) containing c such that f(x) > 0 for all z € (a,…

A:

Q: For each of the following sequences, find the limit points. For each limit poir quence that…

A: In this question, the concept of convergence is applied. Convergent The feature of approaching a…