3. Suppose A and B are non-empty sets of real numbers that are both bounded above....

3. Suppose A and B are non-empty sets of real numbers that are both bounded above.

(a) Prove that, if A ⊆ B, then supA ≤ supB.

(b) Prove that supA∪B = max{supA,supB}.

(c) Prove that, if A∩B 6= ∅, then supA∩B ≤ min{supA,supB}. Give an example to show that equality need not hold.

Expert Solution

Colby Messinger answered 1 year ago

Let A and B be two non empty bounded subsets of R: 1) Let A +B...

Let A and B be two non empty bounded subsets of R: 1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B 2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)

Let A and B be sets of real numbers such that A ⊂ B. Find a...

Let A and B be sets of real numbers such that A ⊂ B. Find a relation among inf A, inf B, sup A, and sup B. Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among sup A, sup B, and sup C. Let A and B be sets of real numbers and write C = A ∩ B. Find a relation among sup A, sup B, and sup...

If a set K that is a subset of the real numbers is closed and bounded,...

If a set K that is a subset of the real numbers is closed and bounded, then it is compact.

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only...

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only if every infinite subset of E has a point of accumulation that belongs to E. Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real numbers is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E. Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Suppose that a subset S of an ordered field F is not bounded above in F....

Suppose that a subset S of an ordered field F is not bounded above in F. Let T be a subset of F satisfying the property that, for each x ∈ S, there exists y ∈ T such that x ≤ y. Prove that T is not bounded above in F.

Let S be the set of natural numbers which can be written as a non-empty string...

Let S be the set of natural numbers which can be written as a non-empty string of ones followed by a non-empty string of zeroes. For example, 10, 111100 and 11100000 are all in S, but 11 and 1110011 are not in S. Prove that there exists a natural number n∈S, such that 2018 | n.

. Let U be a non-empty set. For A and B subsets of U, define the...

. Let U be a non-empty set. For A and B subsets of U, define the relation A R B if an only if A is a proper subest of B. a. Is R reflexive? Prove or explain why not. b. Is R symmetric? Prove or explain why not c. Is R transitive? Prove or explain why not. d. Is R antisymmetric? Prove or explain why not. e. Is R an equivalence relation? Prove or explain why no

A, B and C be sets. (a) Suppose that A ⊆ B and B ⊆ C....

A, B and C be sets. (a) Suppose that A ⊆ B and B ⊆ C. Does this mean that A ⊆ C? Prove your answer. Hint: to prove that A ⊆ C you must prove the implication, “for all x, if x ∈ A then x ∈ C.” (b) Suppose that A ∈ B and B ∈ C. Does this mean that A ∈ C? Give an example to prove that this does NOT always happen (and explain why...

Suppose the sets A and B have both n elements. 1. Find the number of one-to-one...

Suppose the sets A and B have both n elements. 1. Find the number of one-to-one functions from A to B. 2. Find the number of functions from A onto B. 3. Find the number of one-to-one correspondences from A to B.

Suppose the government sets a price floor that is above the equilibrium price for a given...

Suppose the government sets a price floor that is above the equilibrium price for a given good. d)It can be said that at the price floor, a)although sellers are selling all of the product that they desire at this price, the consumers are not able to buy all that they desire. b)although consumers are purchasing all of the product that they desire at this price, the sellers are not selling all that they desire. c)both sellers and buyers are satisfied...

Question