Assume all sets are subsets of a universal set U. Suppose that A, B, and C are arbitrary sets, Prove that: (A – C)n (B – C) = A – (B° U C) (a) by using element argument. (b) by applying the algebraic proof method.

Q: Consider R×R to be a universal set with subsets A, B, and C defined as follows. = {(x,y)x* + y° s1}…

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Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: Consider  the relation R={(a, b),(a, c),(c, b),(b, c)} on  the set A={a, b, c}. Is R reflexive? Is R…

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Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Let R be the set of real numbers equipped with the standard topology. Let A and B be two subsets of…

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Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

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Q: Let R be the set of real numbers equipped with the standard topology. Let A and B be two subsets of…

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Q: Use Induction to prove for any non-empty set A and any partition of the set B by non-empty sets…

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Q: Let E CR be a closed set. Prove that aE CE.

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Q: Let X, Y be two sets. Prove that X × Y C P(P(X U Y )).

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Q: Let B= {x €R|x E Q} be a set. Is B countable? Justify the answer by showing the explicit formula for…

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Let X, Y be two sets. Prove that X × YC P(P(X U Y )).

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: Let A and B denote subsets of R". (a) Prove that ANBCANB. Give an example to show that equality does…

A: (a) We know that for any two sets A, B A∩B⊆A⇒A∩B¯⊆A¯                        (1) ( Since if X⊆Y then,…

Q: Give an example of a compact set A and a closed set B in R2 such that (conv A) N (conv B) = Ø but A…

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Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: 2. For all sets X and Y, prove the given statement by the indicated method. (a). Method of algebra:…

A: The statement to be proved  is: Xc∪Xc-Yc=X Take left hand side of the statement and evaluate using…

Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: Prove that: m*(A) 2 0, for all sets A

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Q: Prove that if S is a bounded set then S is bounded.

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Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Both P and Q are transitive relations on set X. Proof that a. Pn Q is also transitive b. PoQ is also…

A: Given,    Both P and Q are transitive relation on set X.

Q: Given an infinite set X, construct a bijection from X to a proper subset of X.

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Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: Let X be an infinite set, then any topology on X is also infinite. Select one: O True O False

A: Note: " Since you have asked multiple question. As per our guidelines we are supposed to solve only…

Q: ( Prove that the set {a E R2: a2 ... Prove that the set {a E R2: a2 not equal to 6} is open.

A: Let the set is A=a1,a2∈ℝ2:a2≠6 The objective is to prove that the set A is open.

Q: Let A and B be sets. Fill in the blanks to define precisely what it means for A to be a subset of B.…

A: Let A and B be sets. To fill in the blanks to define precisely what it means for A to be a subset of…

Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

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Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: For nonempty sets A, B, and C, let f: A > B and g : B >C. Prove: If g o f is one-to-one, then fis…

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Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: T have to show rigorous proof of the inclus (An B) x C c (Ax C)UCBXC) 1,B.and C, Sets non-em pty of…

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Q: Let X be a set with an operation *:X² → X such that (i) x * x = for all x E X, %3D (ii) x * (y * z)…

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Q: Let A, B, C be sets. Prove that if f: A → B and g: B → C are bijections, the composition gof: A → C…

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Q: Let A, B be disjoint infinite sets. Prove that their union A U B is denumerable if and only if both…

A: Given A , B be disjoint infinite sets. Firstly, consider A∪B is denumerable Since, A⊆A∪BandB⊆A∪B…

Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

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Q: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

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Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: 2. Let T={USR: U = Ø or U= R or U=(-∞o,a) for some a E I}. Prove that T is a topology on R (set of…

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