3. Suppose a, nEZ and 0 < a< n. Prove that - u 1 + a a (:) in terms of factorials; and secondly by consid- in two different ways: firstly using the formula for ering subsets of {1, ..., n}.

Q: Suppose that there are nine students in a discrete mathematics class at a small college a) show that…

A: Given: 9 students in a discrete mathematics

Q: Use direct proof to show, "If m + n and n + p are even integers (s and t), where m, n, and pare…

A: To answer the given questions we shall use the following definition  An integer n is said to be even…

Q: Let (A, 3) be a poset, and let BC A. Prove that B has at most one greatest lower bound.

A: Let, A,&lt;~ be a poset and B is a subset of A and a∈A. Then by the definition, a is the greatest…

Q: Show that if a and b are positive integers, then there are divisors c of a and d of b with (c, d) =…

A: image is attached

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: with N=10 so set Of dah lir a popunhon bunoling 160 b) Calcolale the popuhhon Jhndard dernhon.

A: Formula :

Q: Suppose that S is a subset of N and there is a natural number b such that if m E S, then m < b. Show…

A:

Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: Represent in the Argand Diagram all such z for which |z + 13| < 15

A:

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L,=u, + 2u,-1

A: Introduction: We know that the relation of Fibocanci number and Lucas numbers is given by  F0=0,…

Q: 2. Suppose a, b, and care integers. Prove that if a |band alethen a |b+c and a|b-c.

A: Assume a|b and a|c.

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: 7) Show that for any permutations a and ß in Sn, a*ß and ß have the same parity.

A: We have to show that a4b and b have same parity in the group.

Q: Suppose that a +b+c= d.where a, b, c, d are nonnegative integers. Let P be the assertion that d is…

A: We will use basic logic theory to solve this question using parity argument.

Q: 4. Consider the following axioms: a. Every bird sleep in some tree. b. Every loon is a bird, and…

A: as given (a) every bird sleep in some tree. (b) every loon is a bird , and every loon is aquatic.…

Q: Suppose a, n E Z and 0 < a< n. Prove that ()- (":")- (:-) n - 1 %3D + a a а- 1 in two different…

A: Given a,n∈ℤ such that 0&lt;a&lt;n. (1) Consider:…

Q: 42 Show that E a Imod n) for all integer aif and only if n is squarefree. Ep(n) is Euler Totient…

A:

Q: 4. Explain why if p is prime then Z, is the only subdomain of Z,

A: To prove:If p is prime, Zp is an subdomain(internal domain).

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: b) Use generating functions to determine how many four element subsets of S = {1,2,3, ..,15}…

A:

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: a. Suppose that we have a class with 30 students. Prove that at least two students will have been…

A: a. Suppose that we have a class with 30 students.Prove that at least two students will have been…

Q: 5. Use the diagram below to prove MOOP P. Q

A: Given the diagram

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

A:

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

Q: п(п + 1) Show that , k 2 k=1

A:

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

A:

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: 8. Show that, at the n stage of the construction of the Cantor Middle- Thirds set, the sum of the…

A: Cantor Middle-thirds set: It is created by repeatedly deleting the open middle thirds of a set of a…

Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

A:

Q: 3 S how that Set M with n elements. a where n is a elem ent of ne tural numbers (ne IN. has 2…

A: Number of elements in M is n. To prove power set of M has 2n elements. We know that Power set of M…

Q: [Q1] From the letters {A, B, C, D, E, F, G}, how many: (a) 3-distinct-letters co can be generated?…

A: Given: There are 7 letters A, B, C, D, E, F, G

Q: Let n and m be integers. Prove that if m + n >= 100, then either m >= 50, or n >= 50.

A: Prove the statement by proving the contrapositive. The contrapositive of the given statement is as…

Q: A. The number of permutations of length n that avoid 132 and have the additional property that their…

A: Based on the additional property, the first entry is larger than the second entry is Cn-1, (the…

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L, = un + 2un-1 for all n…

A: Remark: We know that both the Fibonacci numbers and  Lucas numbers are obtained as the sum of its…

Q: 2. Prove thatN is the smallest inductive subset of R:

A: We can prove this by proving property of induction

Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

A:

Q: Consider a group of n warehouses in a shipping network, where n>=2. In order to ship products across…

A: Here in the question it is given the number of warehouses in a shipping network having a set limit…

Q: Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number p…

A: Every integer greater than 1 can be written as the product of prime numbers. In other words, every…

Q: Prove directly from the definitions that for all integers the Same parity Divide Bnt 19. and E m and…

A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

A:

Q: Prove the following using the appropriate method.    Let a, b be integers. If ab is even, then at…

A:

Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

A:

Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: As we know that any truples h,i,j,k,m with n≥h≥i≥j≥k≥m≥1  can be represented as a string of n-1…

Q: 1.Suppose that C is generated from a set B= {a,b} by the binary operation f and the unit operation g…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

A: As we know that, 1&gt;1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…