iii. Is the set compact in ²? {(a₁, a2, a3, ...) € l² : |an| ≤ = } and <==}

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: Let A and B be sets. Prove that A U B = An B if and only if A = B

A: First show if A∪B=A∩B then A=B, then show that if A=B then we have A∪B=A∩B.

Q: 4. Prove the following distributive law for finitely many sets: An (B₁ U B₂ U B3 U ... U Bn) = (An…

A: We have to prove distributive law for finitely many sets: A∩B1∪B2∪B3∪....∪Bn=A∩B1∪A∩B2∪....∪A∩Bn.

Q: Let A and B be finite sets with A - B = {a, b, c, d, e, f, g} and An B = {h, i, j, k, l, m}. Find…

A:

Q: Assume that a and b are real numbers such that a <b. Prove that the interval (a,b) is an open set.…

A: Given that a and b are real numbers and a&lt;b. Prove that (a,b) is an open set

Q: When is a set compact?

A:

Q: d) Give a recursive definition of the sequence {a„}, n = 1, 2, 3... %3D n if a,= 2 + 1; and a15 e)…

A: d)  Here, given that an=2+1=3 So, for all n, an=3 which is the recursive definition.

Q: Let A, B, and C be sets. Prove (AUB)NC= (ANC) U(BNC)

A:

Q: 4) Let a < b be real numbers. Prove that the half open/half closed interval (a, b) is both an Fo and…

A:

Q: et A, B, C and D be sets. Prove that (AnB)u(CnD)=(AUC) n(BUC) n(AUD)n(BUD)

A: The set A∪B contains all the elements that are present either in set A or in set B. The set A∩B…

Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

Q: 1) Let the set S be defined as S = {x | x is a U.S. President} Find n(S). 2) Define compliment to a…

A: use the basic property of sets .

Q: Which sets are compact? The union of [0, 1] and [2, 3]

A: Concepts introduction: Theorem (Heine-Borel Theorem): A subset S of R is compact if and only if S is…

Q: Consider following five sets: I.{ , } { } II. { } III. IV. { , , } V. { , } Find all set(s)…

A:

Q: Describe the set {TER²: doo (r,(0,0)) = 1}.

A: Given: Set x∈R2:d∞(x,(0,0))=1 We have to describe the set.

Q: Is the set {(a₁, A2, A3, ...) € l² : |an| ≤ compact in 2? √n

A: As per the question we are given a set consisting of the infinite sequence : (a1 , a2 , a3 , ..., )…

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

Q: 2. Prove for any sets A, B that (An B) = A if and only if A C B.

A:

Q: II. Let Q-set of all real numbers, A-index set. Identify the generalized union and intersection of…

A:

Q: Prove T = { ... , −56, −42, −28, −14, 0, 14, 28, 42, ... } is a countable set

A:

Q: Suppose A={-1, 0, 4}.Find a set B such that A is equivalent to B.

A: Two sets A and B are said to be equivalent to each other if both the sets A and B have same number…

Q: Suppose A, B, C are sets. Prove that (AUB) – C = (A − C) U (B − C)

A: A∪B-C=A-C∪B-C

Q: 2. Find a set A such that A = {2, JA]}.

A:

Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

A:

Q: (i) If A and B are two sets such that ACB, then m* (A) ≤m* (B).

A:

Q: Show that the interval [0,1] is a closed set in R

A:

Q: VII.1. Describe the set (R × Z) ∩ (Z × R)

A: We have to describe the set (R × Z) ∩ (Z × R).

Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

A:

Q: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

A:

Q: 11. For sets A and B, define AAB=(A\B) U (B\A). (a) Let A and B be sets. Prove that A = B if and…

A: For sets  A and B , define                             A∆B=(A \ B)∪(B \ A) .  Then we have to show…

Q: Consider the following sets: U= {−5, −4, −3, −2, −1,0,1,2,3,4,5}, A = {x : x^2 = 25}, B = {−5,…

A: Note: Hi, thank you for the question. As per our company guideline we are supposed to answer only…