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: Use the element method and contradiction to prove that for all sets S and T S Ռ ( T - S) = {  }

A:

Q: Let A, B, C, D be sets. Prove that (a) A × (BN C) = (A × B) N (A × C') (b) (A × B) N (C × D) = (AnC)…

A: Given (a) A×B∩C=A×B∩A×C (b) A×B∩C×D=A∩C×B∩D

Q: The greatest common divisor (gcd) of non-zero integers x and y is the the smallest positive integer…

A:

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

A:

Q: 2 > ]x – 4| X-1

A: Given,          2x-1≥x-4

Q: (2) Is the set {r € Rr² € N} countable? Justify your answer.

A: Yes the set A={x €R : x2 €N } is countable

Q: Q.3 Explain operations on classical / crisp sets

A: There are various operations on Classical sets.   The first kind of operation is Union. The union of…

Q: Prove that the K4 is not consecutive.

A:

Q: In cartesian product, prove that (A-B)XC=(AXC)-(BXC) A,B,C are sets

A:

Q: etermine whether the set of x-2x²,2+5x-x²x+x²} is linerly independent in P

A: First we use definition of linearly independent Set and we check given vectors are linearly…

Q: Let A = {1,2} and B = {3, 4}. (a) Find A x B (b) How many subsets Ax B of size 2 are there? List…

A: Given that A= {1, 2} and B= {3, 4}A x B is the cartesian product

Q: Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that(i) A × (B ∩ C) = (A ×…

A: i) Answer(i) A × (B ∩ C) = (A × B) ∩ (A × C) (proved)

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: R:= {(a, b) | a < b} R:= {(a, b) | a = b} Rs:= {(a, b) | a =b or a =-b} R:= {(a, b) | a = b+1} Rs:=…

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: Is a R a maximal deal

A: No, it is not necessarily true that aR is a maximal ideal in a commutative ring R with unity. In…

Q: Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C') = (A × B) \ (A × C). Prove that (A x B) n (C…

A:

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) Use an element proof to prove that for all sets A, B, C, and D, if An C =0 then (A x B)n (C x D)…

A: The proof of the given statement is given in the explanation part.Explanation:Steps and explanations…

Q: Does A − (B − C) = (A − B) ∪ C hold for all sets A, B, and C? Prove or give a counterexample for…

A:

Q: 1. (a) Let A = {k, k +1,..., k2}, for k = 1,2,3,.... Uº1 Ak - n=1 Ak (b) Find two infinite sets A…

A: Part (a):- Given that  Ak =  k, k+1, ......, k2 ,   for  k=1, 2, 3,....Find    ∪k=150 Ak - ∩k=1∞ Ak…

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 P(A₁) > 0 and if A2, A3, A4,... are mutually disjoint sets, show that P(A2 U A3 U ... |A₁) =…

A:

Q: Let X be a set (P(X),Δ,∩) where  P(X) is the power set of X, and Δ is the symmetric difference. Does…

A:

Q: Show that the following statement is false. If A and B are two sets of positive integers, then…

A: Let A={2,3,4}    B={5,6,7} U={1,2,3,4,5,6,7} Now, A'={ 1,5,6,7}  B' ={1,2,3,4} Now,…

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: Determine the orders of the elements of D∞

A: It is known that, D∞=a,b|a2=b2=e also it can be observed that, D∞=abka,abk,bak,bakb Let x∈D∞. If…

Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Cartesian product and identity

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

A:

Q: or each of the following structures, determine whether the set of all integers divisible by 3 is…

A: We the set of all integers divisible by 3 is S={±3,±6,±9,...} For an element to be ϕ-definable in…

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: 3. Let V be the set of all ordered triple in R³, and F, the set of real numbers, where the…

A:

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: (10) Prove that P(A)n P(B) is equal to P(An B) for any sets A and B. What can we say if the…

A:

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

A:

Q: In the following list, select all the correct statements, if (4,) is a poset and for any elements…

A:

Q: 1. List a pair of zero divisors of (M₂(Z), +, ×):

A: