Suppose that A is a set of 10 natural numbers, all of which are less than 50. Show that A must have at least 2 subsets, the sums of whose entries must be the same.

Q: Let S be a set of non-negative integers with 80 elements. Show that we can find a subset of S such…

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Q: Q2 Let B = {1,2,3, ..., 1240}. Prove that in any subset of 621 elements of B, there are two elements…

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Q: Show that in a sequence of m integers there exists one or more consecutive terms with a sum…

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Q: Prove or disprove that given a set of 125 integers, one can choose a subset of S of 16 of them so…

A: Given we have set which contains 125 different numbers. When we divide each number in set by 8 we…

Q: 8. Consider a sequence {Cn} of nonempty compact subsets of R such that Cn+1 C Cn for each n > 1.…

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Q: Suppose that every student in a discrete mathematics class of 31 students is a freshman, a…

A: Given that, every student in a discrete mathematics class of 31 students is a freshman, a sophomore,…

Q: Let x be the set of all positive integers and d(min) = | = - = | show that (x,d) is not complete

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Q: Let n and k be natural numbers with n >k. how many subsets of {1,2, ..., 2n} have exactly k even…

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Q: Let A be a set of m elements, where m is a nonnegative integer. Suppose you will prove by induction…

A: PRINCIPLE OF MATHEMATICAL INDUCTION: It is principle which is used to prove a statement or theorem.…

Q: 12.32. Give an example of a set S of four (distinct) positive integers such that the greatest common…

A: We have to find the required set S with the above property.

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: Show that for any positive integer n, with n not equal to 2, there exists a partition of n that is…

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: Given 10 numbers a1, a2,.… , a10, from the set S natural numbers S = {1,2,3, ., 100} . Prove that…

A: Consider the set S=1, 2, 3, …, 100. Let a1, a2, …, a10 be 10 numbers from the set S. Take the set…

Q: Let a1, a2, a3, ... be the sequence of numbers defined by a1 = 1, an = V2+ an-1 if n > 2. Prove by…

A: Given that  a1, a2, a3, ...... be the sequence of numbers defined by a1=1, an=2+an-1  if n≥2. Then…

Q: Let a, b be integers for which a01 and ba02 are relatively prime. Show that a and b are also…

A: Given: a and b are two integers for which a101 and b202 are relatively prime. We have to show that a…

Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: . Let m be any odd whole number. Then m is a divisor of the sum of any m consecutive whole numbers.…

A: I have supposed that m is any odd whole number and then I have taken another arbitrary whole number…

Q: Find the sum of the positive even integers up to and including 50.

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Q: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

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

A: We used binomial inequality

Q: Every none-empty subset of integer numbers contains a least element.

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

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Q: 5. Exercise $1.3 #50. Prove that every integer greater than 6 can be expressed as the sum of two…

A: To prove - Every integer greater than 6 can be expressed as the sum of two relatively prime integers…

Q: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

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

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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: Prove that the set of all 2-element subsets of N is denumerable.

A: According to the given information, it is required to prove that set of all 2 elements subset of N…

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

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Q: 6. (a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the…

A: (a) Show that any set of 5 integers at least have two the same remainder when it is divided by 4…

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: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: 4. Prove that for every n > 1,3 divides n³ – n.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

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

A: The number of r combinations which can be selected from n numbers if repetitions allowed is…

Q: Show that the set of numbers of the form k/5", where k is an integer and n a positive integer, is…

A: The set of numbers of the form k5n where k is an integer are dense if every real number has rational…

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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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: Find all positive integers n such that p(n)|n.

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Q: Use mathematical induction to prove that for every integer n ≥ 2, if a set S has n elements, then…

A: To prove: For every integer n ≥ 2, if a set S has n elements, then the number of subsets of S with…

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

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Q: 7. Prove: If A is any set of 9 integers between 1 and 50, then there exist two different subsets XC…

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Q: Consider the infinite set Yof all infinite strings of binary numbers 0,1, for example…

A: We will first assume that Y is countable, then we will use a clever method to construct an unique…

Q: 11. Which of these collections of subsets are partitions of the set of integers? (a) the set of even…

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Q: Consider an arbitrary set S of 10 integers on the range [0, 100]. Prove that there exist two…

A: Discrete mathematics is majorly applied on those sets which are discrete in nature. Its study is…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.