Let n be a positive integer and X = {1,. X even if Y is even and odd if | Y| is odd. i. How many subsets does X have? 2n+1}. We will call a subset Y of ii. Prove that X has the same number of subsets of size n as subsets of size n + 1. iii. State and prove a formula for the number of even subsets of X. Hint: Find a bijection between the collection of even subsets and the collection of odd subsets.

Q: 13. Recall Z = {..,-2,–1,0,1,2,..} (the integers). Let Z* = {1,2,3,.} be the positive integers. Let…

A: Given: ℤ=…, -2, -1, 0, 1, 2, … and ℤ+=1, 2, 3, …

Q: Give a direct proof of the theorem: “if x and y are odd integers, then x + y is even.”  DISCRETE…

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Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: Let R be the set of all real numbers. Is R*, R , {0}} a partition of R? Why or why not? Select- v.…

A:

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Find the number of ordered pairs (m,n) such that m and n are positive integers in the set {1, 2,…

A: Let us consider Tm=2m+1.  m Tm 1 3 2 5 3 9 4 17 5 33 6 65 7 129 8 257

Q: of subsets for M = {x | x is an integer between 1 and 7} Set M has subsets.

A: We have to check whether given statement is true or false.

Q: Determine whether the statement is true or false. Prove the statement directly from the definitions…

A: a. For all integers a,b, and c, if a | bc, then a | b or a | c.To demonstrate this assertion, we'll…

Q: Determine whether each statement below is true forn = 1, 2, and 3. Enter a comma-separated list of…

A:

Q: Find the number of subsets for the set. M = {x|x is an integer between 4 and 11} Set M has subsets.

A:

Q: Prove bey wnduc hiou that for n22.

A:

Q: Challenge. Establish the Cauchy-Schwartz inequality: For real numbers we have (a + az + ..+ a) (b+…

A: Prove the Cauchy-Schwartz inequality…

Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: Suppose that x is a real number and that UCR. Prove that the following two conditions are…

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Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: Find the complement of the set given that  U = {x | x I and −3 ≤ x ≤ 7}. (Enter your answers as a…

A: First we write U in roster form.

Q: Consider an indexed family of sets. B={Bn:neN}, where Bn = {2n, 2n + 1, 2n+2, ..., 3n}. (a) Write…

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