Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper bound of E. Prove that a

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: Show for every nonempty, finite set E that supE = maxE.

A: Given, E is a finite set. E is a non-empty set. To prove, supE = maxE We know that, sup is supremum…

Q: If E <R, then for any 0 <E or an open set there exists b such that m * (G – E) < ɛ. Give an example…

A: For any non measurable set E, the above relation does not hold. The Example of such non measurable…

Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: Find a ≤ b such that

A: ∫abx2-9dxApply the Sum Rule:…

Q: Suppose that X is a set and {A. : a E I} is an indexed family of sets (which contains at least one…

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

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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: Prove that the set Q \ Z is countable.

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Q: Find the value of z so that -8' 1 -4 is in the set M= span 12 2 2 Enter the value below:

A: Let's find.

Q: Let A and B be finite sets such that B = {b1, b2}. Prove that |A – B| < |A|.

A: We are given that A and B are two finite sets, where B= {b1, b2}.We have, |B| = 2, and let |A| =…

Q: dentify the least upper bound and greatest lower bound of the following sets. If it cannot be…

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Q: Suppose that A is a nonempty set of real numbers, and suppose that a ER is an upper bound for A.…

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Q: Let A and B be nonempty sets. Prove that A x B = B x A if and only if A = B.

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Q: Let A be an uncountable set, and B a countable set. Prove that AUB is uncountable.

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Q: Let X be a set. Show that P(X)÷ X.

A: Solution:Let X ba a set.To Prove: P(X)≠X

Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

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

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Q: Suppose E is a bounded, nonvoid set of real numbers. Define |E| to be the set { x: x = E}. Is it…

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Q: 4. Let C₁ be the set of all continuous real-valued functions on I = [0, 1]. (a) Show that d : C₂ ×…

A: According to our guidelines I can answer only Question no. 4(a).

Q: Show that; is a linearly independent set in R3.

A: S=121,011,201 Let a,b and c be the scalar in R3. Then write the set in term of linear span.…

Q: Let E be a set of real numbers. Show that x is not an upper bound of E if and only if there exists a…

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Q: Show that the translate of an Fδ set is also Fδ.

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Q: Exercise 1.4 Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E, and 3 is…

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Q: Find all topologies for the set X={a,b,c}. Justify your answer.

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