Prove that if A1 and A2 are compact sets then A1 U A2 is compact.

Q: Let A, B and C are sets. Prove that <= is transitive

A: For proving transitivity we need to show that if A&lt;=B and B&lt;=C then A&lt;=C

Q: 4. Prove that if A and B are convex sets, then AN B is also convex.

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Q: Prove that if A and B are transitive sets, then A ∪ B and A ∩ B are transitive sets

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Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Let S and T be sets. Prove that SUT = T iff S is a subset of T.

A: To Prove S∪T=T iff S is a subset of T

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: Prove that if A and B are sets, then P(A) U P (B) C P(AUB).

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Q: 3. Let {B;}je1 and {C;};e1 each be a family of mutually disjoint sets. If B; = C; for each i e I,…

A: Answer :

Q: Let A and B be sets and suppose that A U B = B. Prove that ACB.

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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: Prove that for sets  A and B,  we have  (A∪B)−(A∩B)=A−B if and only if B⊂A.

A: By theorem we know that,            A⊂B if and only if A∩B =A and  A∪B =B

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

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Q: Prove that if A is transitive set, then A+ is also a transitive set.

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Q: Let A and B be two subsets of E. Prove using the set identities: А - В — В — А.

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Use an element argument to prove that for all sets A and B, (An B) U (AN BC) = A. %3D

A: I have proved that sets on both side are subsets of each other and ten concluded that they are equal

Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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

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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: Can a set and it's Complement be both nowhere dense in R? A Ac=R-A

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

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Q: Prove formally that A\ B= A\(An B), for any sets A and B.

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Q: Prove that for all sets A and B, A – B = AN B

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Q: If sets X and Y are countable, then prove or disprove that X-Y is uncountable.

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Determine whether the following theorem is true or false: For all sets A, B and C, if A is a subset…

A: If X is a strict subset of Y, but X is not equal to Y which means there exists at least one element…

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

A: Identities of the sets

Q: Prove that if A and B are sets, then (A∩B)∪(A∩B^' )=A

A: To prove : A∩B∪A∩B'=A   Let x∈A∩B∪A∩B' ⇒x∈A∩B or x∈A∩B' ⇒x∈A and x∈Bor x∈A and x∈B' In either of the…

Q: Prove that if A is a set then so is {A}!

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Q: Write up a formal proof of the following: Claim: For any set A, A ⊆ A.

A: To prove rigorously that any set A is a subset of itself.

Q: 1. Prove that, if A, B, and C are sets and if ACB and ACC, then ACBOC.

A: To prove: A⊆B∩C

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

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: O No 4- Let x bea and That sad anly Aeplegical Space sibspace fxiShow chred a subset' C of y is if…

A: Y be a subspace of X means that any open set in Y is the intersection open set in X with Y

Q: Suppose A, B, and C are sets. Prove that (AU B) – C C AU (B – C). - -

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Q: 1. Prove that a set SCR is everywhere dense if and only if S intersects every non-empty open set in…

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Q: Let Ω be a set and A = {A ⊂ Ω : A is finite or Ac is finite }

A: given that Ω is a set and also given A ⊂Ω if Ω contains n elements  then A contains (n-1) elements…

Q: Let A,B be sets. Prove that AAB = if and only if A= B

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Q: 1. Prove the following statements about sets A and B. A B= AN Bº %3D

A: Given: Let A and B be two sets. To prove,A-B=A∩Bc

Q: Let A and B be two finite sets such that |A| = 20, |B| = 28 and |A U BI

A: Given A=20B=28A∪B=36

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: Let A be a nonempty subset of a bounded set B. Why does inf A and sup up A exist? Show that (a) inf…

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

A: Solution:-

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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Q: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

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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Average

In mathematics, everyone must have heard the term average in one place or other. Almost all types of calculation require taking out an average. In everyday life, the term is used for denoting a certain amount that can be said to be a typical quantity for multiple people or objects. In other words, if a group of people possesses a certain object, then the average gives us an idea regarding the quantity of the object possessed by each person. However, this is solely based on mathematical calculation. It is important to note that most people will not strictly have the average value in a group.