A subset S of the integers {1, 2, ..., n} is called extraordinary provided that its smallest integer equals its size: min{x : x ∈ S} = |S|. For example, S = {3, 10, 17} is extraordinary. Let gn be the number of extraordinary subsets of {1, 2, ..., n}. Prove that with g_1 =1and g_2 =1. g_n=g_(n−1)+g_(n−2), fo rn≥3

Q: G={a+bi: aE Z, bEZ , i =(-1)^(1/2)} Show that the set of Gaussian Integers has the same power as…

A: Given that G={a+ib : a,b are integer and i=-1}  We have To show that G has same power as N.

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

A: Using division algorithm.

Q: ccording to the National Center for Health Statistics, 15% of all Americans have hearing trouble. In…

A:

Q: Consider the following families of sets indexed by the set of natural numbers. Which among these is…

A: Given: Intersection or union of a family of sets. To find: Which of the intersection or union of a…

Q: 10. Prove that (n) ≤ n/2 for n ≥ 8.

A: To prove that π(n) ≤ n/2 for n ≥ 8, where π(n) represents the number of prime numbers less than or…

Q: Suppose that the number of subsets with k elements, made out of a set of n elements is called by the…

A:

Q: What does the K stand for in Kn? (Finite)

A:

Q: If there are a total of 8 chocolates be given to to three children such that each children receives…

A: Given: 8 chocolates  are to  given  to 3 children such that  each children receieves at least two…

Q: 10. Show that the interval (0, 1) is covered by the open intervals In (1/n, 1- 1/n), n 2 3, but (0,…

A:

Q: Prove that the following functions are computable. n-m if nm (a) -:IN² →→ IN defined by n- m:= 0 (b)…

A:

Q: Assume n and m are two integers such that n > m > 1. If a bit string of length n+m is randomly…

A:

Q: Let f= (14723 5 6) and g = (1 5 3 264) be permutations in S. What is the second line of fin two-line…

A: Introduction: Permutation businesses are mathematical businesses whose individuals are permutations…

Q: Answer the following questions for the poset ([[1]. [2), (4), (1, 2), (1, 4), (2, 4), (3, 4), (1, 3,…

A:

Q: x=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14] x61=7 mod15 find x in range

A:

Q: (3) Find the formula for the number of ways for putting n identical balls into k distinct boxes so…

A:

Q: Prove that If a is an element of order n and p is prime to n, then a\power{p} is also of order n.

A: is prime to We have to show

Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

A:

Q: An urn contains m number of red balls and n number of white balls. Two balls are selected without…

A: Solution-: We have given there are 'm' number of red balls and 'n' number of white balls We want to…

Q: Find the least odd composite integer n such that the set T = {2¹, 22,..., 20(n)} is not an RRS…

A:

Q: A positive integer n is said to be square-free if it is a finite product of distinct primes. We also…

A: Given that, μn=-1ωn     if n is square free0                 otherwise That is, μ1=1 and for n&gt;1,…

Q: Suppose X-Neg Bin(r=2 ,beta=3) a) Evaluate p_0 b) Use the recursive formula to obtain p_1 ,p_2 ,p_3

A: X follows a negative binomial distribution.The pmf of negative binomial distribution:

Q: ↑ Use the following cell phone airport data speeds (Mbps) from a particular network. Find Q₁. 0.1…

A:

Q: n (₁+) (+) () ---- () = 10

A:

Q: prove that there are Hadamard matri X of order 25 and 64

A:

Q: Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

A:

Q: What is the maximum number of people you can invite so that at most 3 guests are born same month?…

A: The Pigeonhole Principle: If n + 1 objects are placed into n boxes, then some box contains at least…

Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

Q: 3. Let F- (A.) be a finite collection of pairwise disjoint finite sets (n) Ume mathematioal…

A: Let F=Aii=1n be a finite collection of pairwise disjoint finite sets. (a) Prove ∪i=1nAi is finite…

Q: How many ways are there to distribute r+s identical balls among five different children such that…

A:

Q: Exercise 8: Let x ER\Z, and suppose x has a finite number of decimal places. Let n be the numbe of…

A:

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.