Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

Q: Find the sup and inf of the following set ∩(1 − 1/n, 2 + 1/n) where n ∈ N

A: The problem is asking us to find the supremum (least upper bound) and infimum (greatest lower bound)…

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

Q: Find the intersection between the set of all integers bigger than 0 but smaller than 10 and the set…

A:

Q: For all integers a, b, and c, if a/b and a|c, then a|(5b – 7c). -

A:

Q: Show that 2N, the collection of all sets of natural numbers, is uncountable.

A:

Q: Prove by Contradiction that √6 + √8<√2

A:

Q: In each of (a)-(f), answer the following questions: Is AS B? Is B C A? Is either A or B a proper…

A: As per the policy of bartleby the provisions is to solve only one 3 subparts of questions at a time

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

A:

Q: 5. Let r,ne N. Show that the product (n+1)(n+2)...(n+r) of r consecutive positive integers is…

A:

Q: The coefficient of x³ in (1 + x² + x² + x6)4 is determined by the 4-tuples o integers (a1, a2, a3,…

A:

Q: 6.) Let A = {small, medium, large, extra large}, B = {purple, C= {t-shirt, polo, hoodie}. a. List 2…

A: Definition : A×B×C=a,b,c|a∈A,b∈B,c∈C We have  A={small, medium, large, extra large} B={purple, gold}…

Q: Prove that 17Z ∪ 7Z is countabl

A: A set X is said to be countable if there exists a bijection between f: X -&gt; N, where N is the set…

Q: Suppose that each of A, B, and C is a set and each of BnC, A–(BnC), A-B, A-C, and (A – B) n (A – C)…

A:

Q: 3. We say that an integer c is a common divisor of integers a and b if cla and cb. We say c is the…

A:

Q: If A is a perfect subset of B and P(a < P(b)), then P(B - A) is equal to Select one: O A. P(b) -…

A: Given that,  A is Perfect subset of B and, P(a &lt; p(b))

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: Use the pigeonhole principle to prove that any 6-element subset of {1,2,..., 10} contains a pair of…

A: This question from discrete mathematics and topic is pigenhole principle.

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Denote by mZ the set of all integer multiples of an integer m. More explicitly, тZ 3 {..., -Зт, -…

A:

Q: (ii) Determine supremum and the infimum of the set A=(0,10)∩R∪[−2,8]A=(0,10)∩R∪[−2,8] if they exist

A: It is assumed that R denotes the set of real numbers.

Q: Prove, directly from the definition of divisibility, that for any integer x#0, x|0, 1|x and x|x.

A: Use the following definition of divisibility to prove the required result. Let a and b be any two…

Q: Finish the following set builder notations: The set of even numbers that are also perfect squares…

A: The set builder notation of “The set of even numbers that are also perfect squares where x is an…

Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: Let x € Z. Prove that if x² - 8x + 11 is divisible by 4, the x6- 1 is also divisible by 4.

A:

Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

A:

Q: Consider the following sets. Suppose the variable x is an element of the set of integers. A = (x/x²…

A:

Q: Find ntegers G and t such that gcd (963,652)3967+t-652,

A:

Q: Find all of the units in Pn and verify where Pn is the power set of the set {1,2,...,n} with…

A: Find all of the units in  and verify where is the power set of the set  with addition defined as…

Q: Let U = {x: x is an integer and 2 7}, B = {x: x² > 20}. ii. A = {x: x2 - 3x + 2 = 0}, B = {x:x + 7…

A: We have to determine which condition applies on set A and B.

Q: Suppose a is divisible by b. Show that gcd(a, b) = b.

A: Consider the given information. As we have given that a is divisible by b. then there is a number c…

Q: 5. Find all integers n such that Z6 x Z10 x Z4 has an element of order n.

A: