Let A, B, C be sets. Prove that if f: A → B and g: B → C are bijections, the composition gof: A → C is also a bijection.

Q: Prove that for all sets A, B and C we have the identity (A\B)∩C =(A∩C)\(B∩C).

A: See the attachment

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

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Q: LAUBUC = ANĒNT. %3D

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Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

A: Since you have asked multiple questions, we will solve the first question for you. If you want 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 A, B sets such that A ¢ B and B C C. Prove that A ¢ C.

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Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

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

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Q: Let C, D be subsets of a set U. By proving two inclusions, show that (CC n D) U (C n D) = D where Ce…

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Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

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: Prove that the class {AC – B, BC – A, AB − C} is pairwise disjoint.

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: If C1, C2 are connected subsets of R, then the product C, xC, is a connected subset of R?,

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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: Consider S to be the universal set. Let X = (AUB) nC and Y = (AnT)U (An B). %3D Write out the…

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Q: Prove that for any sets A, B, C, ANB = AN C and AUB= AUC if and only if B = C. %3D Prove that for…

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

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Q: Let A and B be sets. Prove that Ac B if and only if AnB = A.

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

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Q: Prove the following theorem directly. 'Let B, E,W be sets. If B Ɔ W, then BU E WUE"

A: Given B, E and W are three sets .

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: For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B…

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Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

Q: Prove that, for any sets A and B, it is true that A ⊆ B iff A ∩ B = A.

A: Use the following definitions, to obtain the solution. Let A and B be any two sets. The…

Q: Let A = {x ER|a° e Q} be a set. Is A countable? Justify the answer by showing the explicit formula…

A: A is set of all real numbers whose cubes are rational.  Now for any real number b, the cube can be…

Q: 2. Use the element method of proof to prove that for all sets A, B, and C, (An B) -CCANC.

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Q: Let A={(m,n)∣m,n∈N and gcd(m,n)=1} and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc…

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Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

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Q: For any set A and B prove that: a) А - В %3D А - (ВПА) b) BC A' iff A n B = ¢ с) (А — В) U В %3D…

A: The given question can be solved as shown in step 2 and 3. As per the guidelines of bartleby I'm…

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

A: Please check the answer in the next step

Q: Let A and B be sets. Prove or disprove that B – A = BU (AnB)

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Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

Q: Let Cc R be a connected set and a , be R. Prove that [a , b] c C.

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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Q: Let A={(m,n)∣m,n∈N2 and gcd(m,n)=1}, and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc.…

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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: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

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: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

Q: Prove that if A is closed and B is open, then A \ B is closed.

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Q: Let R and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

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Q: 3. Let A and B be arbitrary sets. Use double inclusion to prove that (A U B) N (A U B') = А.

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Q: Prove that only ∅ and R are open subsets of R.

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