C. Think of a set with m + n elements as composed of two parts, one with m elements and the other withn elements. Give a combinatorial argument to show that m+n' ("") = (") (") + (")(,".) + (") (,"2) + ... + (") (") m

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

A: Using division algorithm.

Q: The Boolean sum of the variables z, y and z or their complements, that has the value 0 if and only…

A: Given x=0y=1z=1 Thus yc = 0zc = 0 We know that  a+b = 0 if and only if both a and b are zero

Q: 14. Let (a, b) = 1 and 2 = y for some integers x and y. Then prove that there exists an integer n…

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Q: Suppose (an ) is Cauchy and that for all m, k ≥ 100, |am − a k | 0. Does it follow that the…

A:

Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

A:

Q: п 12. Show that Ë O)=)2"-". Using Combinatorial Proof. k=0

A: To prove: ∑k=0nnkkm=nm2n-m

Q: Use a combinatorial argument to prove the following equality: (C) · k = =n(") k-

A: A combinatorial argument or proof is an argument that involves counting. Here is the proof.

Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

A:

Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

A:

Q: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

A:

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

A:

Q: Compute f2,f3, and f-1 for each of the following permutations. f= (2, 7,4,3,5)

A: Given: f= (2, 7,4,3,5) To Compute: f2,f3, and f-1

Q: 3. Give combinatorial proof of a k (") = n () k-1

A: We know that the combination formula: C(n,r) = n!/(n-r)!r!

Q: Show that gcd (f1,..., fa) = gcd ( f1, gcd (f2, ... , f,)) for any s > 3.

A: The greatest common divisor of two numbers a and b is the number that divides both a and b. It is…

Q: Show that A5 has 24 elements of order 5, 20 elements of order 3, and 15 elements of order 2.

A: We know that A5 has only even order permutation. A5 has 5!2=60 elements. Now write the partition of…

Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: Theorem 1.4.1 Let n(A) denote the number of elements in set 4 n(A, × A2 × ...× Aµ) = n(A) n(A2).…

A:

Q: 4. (a) Write down four distinct elements of the set {2 €C:2- E = 2i}.

A: Note:  We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 3. Suppose a, nEZ and 0 < a< n. Prove that - u 1 + a a (:) in terms of factorials; and secondly by…

A:

Q: (a) Evaluate I = SoSi L' dxdydz. (b) In how many possible orders can you calculate I? Do all give…

A:

Q: 3. Show that if A-1 exists, then eAT dr = A-1(eAt – I) = (eAt – I)A-1.

A:

Q: The_____ of A and B consists of all elements in either A or B or both.

A: The_____ of A and B consists of all elements in either A or B or both.   The answer of the above…

Q: Suppose A1, A2, ..., 430 are thirty sets each having 5 elements and B₁, B2, ..., B₁ are n sets each…

A:

Q: Suppose (an) is Cauchy and that for all m, k ≥ 100, |am − ak| 0. Does it follow that the interval…

A:

Q: Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

A:

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

A:

Q: .

A:

Q: An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of every…

A: Given a n×n magic square

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: (A) Compute the (infinite) discrete convolution (-2, –5, –3, –1) • (–4, –4, –5, –4).

A: Use the sum by column method to calculate the discrete convolution .                 Disclaimer:…

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

A:

Q: Let a1, a2,⋯, a2021 be positive integers. Prove that there exist at least two different sequences…

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: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

Q: Prove that (n!+1, (n+1)!+1)=1

A:

Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

A:

Q: The five most common words appearing in spam emails are shipping!, today!, here!,available, and…

A: It is mentioned that 1 in every 10 messages is spam. Hence, the probability that the message is spam…

Q: Prove that if k <n and (a^k - b^k) | (a^n - b^n), then k |n

A: We will prove it using contradiction

Q: 2. Show that there exists a proper subset A of N (i.e. AC N, but A ‡ N) such that A≈ N.

A: Given that A is a proper subset of ℕ To show: There exists a proper subset A of ℕ which is…

Q: Prove that ℤ/n has exactly n elements.

A:

Q: Let N={1,2,3,….} be the set of all natural numbers. Prove Nx N x N ={(x, y,z)| x,y, and z are all in…

A: Given that N=1,2,3,4,... be the set of all natural numbers. The objective is to prove that…

Q: b) If a collection of objects consists of r, identical objects of type I, r, identical objects of…

A:

Q: Let Σ = {0, 1, +, =} and S = {x = y+z : x,y,z ∈ {0,1}* and x,y,z are binary integers, and x is the…

A: Given that S=x=y+z By using the pumping lemma. Assume that for the purpose of reaching a…

Q: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

A:

Q: [2.1.22] I have 51 rectangular pieces of cardboard, each of which has an integer length and width in…

A: Given: Let's consider I have 51 rectangular pieces of cardboard in set 1,...,100, each with an…

Q: 2. List all the elements of (Z30, Ð) that are of order 15.

A: