Prove the following stronger variant of Proposition 7.4. Suppose C is collection of connected subsets of...

Prove the following stronger variant of Proposition 7.4. Suppose C is collection of connected subsets of a metric space X and B ∈ C. Show, if for each A ∈ C, A ∩ B not equal ∅, then Γ = ∪{C : C ∈ C} is connected. [Suggestion: Consider the collection D = {C ∪ B : C ∈ C}].

Expert Solution

Colby Messinger answered 3 years ago

Prove that the union of a finite collection of compact subsets is compact

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?....

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?. Prove ?(⋂ ?? ?∈? ) ⊂ ⋂ ?(??) ?∈? with equality if ? is one-to-one. 2. Compute: a. ⋂ ∞ ?=1 [?,∞) b. ⋃ ∞ ?=1 [0,2 − 1 /?] c. lim sup ?→∞ (−1 + (−1)^? /?,1 +(−1)^? /?) d. lim inf ?→∞(−1 +(−1)^?/ ?,1 +(−1)^? /?)

Show if A, B, C are connected subsets of X and A∩B not equal ∅ and...

Show if A, B, C are connected subsets of X and A∩B not equal ∅ and B ∩C not equal ∅, then A∪B ∪C is connected.

Suppose we have a collection of n different subsets of the set { 1, 2, ...,...

Suppose we have a collection of n different subsets of the set { 1, 2, ..., n } and they are in some arbitrary order, that is, we have subsets S1, S2, ..., Sn, but how many and which elements are in each of these subsets is entirely arbitrary. Suppose also that we have another subset S' of { 1, 2, ..., n }. (a) Express a brute-force algorithm that determines whether S' equal to one of the subsets in...

Prove this betweenness proposition with justification for each step. If C * A * B and...

Prove this betweenness proposition with justification for each step. If C * A * B and l is the line through A, B, and C, then for every point P lying on l, P either lies on the ray AB or on the opposite ray AC.

Use the laws of propositional logic to prove that the following compound proposition is a tautologies...

Use the laws of propositional logic to prove that the following compound proposition is a tautologies (¬? ∧ (? ∨ ?)) → ?

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection...

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection of vectors in a vector space V and suppose that it is not a linearly independent set. i. Show that one can find a vector w ∈ {v1, . . . , vn} such that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that Span(S) = Span(v1, . . . , vn). ii. Suppose T ⊂ {v1,...

Consider the following variant of theBertrand Model of Duopoly. Suppose there are two firms producing the...

Consider the following variant of theBertrand Model of Duopoly. Suppose there are two firms producing the same good and they simultaneously set prices for their product. If firm i sets a price piand firm j sets a price pj, the total quantity demanded for firm i’s product is given by:qi= 10 –pi+ ½ pjEach firm produces exactly the qidemanded by the market. Bothfirms have the same marginal cost of production: c=4. For example, if a firm produces 5 units it...

For the following exercises, find the number of subsets in each given set. {a, b, c, … , z}

For the following exercises, find the number of subsets in each given set.{a, b, c, … , z}

Prove the following using the properties of regular expressions: (ab)* + c + c* = Λ...

Prove the following using the properties of regular expressions: (ab)* + c + c* = Λ + ab + (ab)* + c(Λ + c*) Λ+ab+abab+ababab(ab)* = Λ + ∅* + (ab)* a(b+c*) + (d+e)* = ab + ac*c* + d + e + (d+e)* a*b + a*a*bc* + d* + ab = d* + ab + a*bc* + Λ a(b+cd*) = a(b+c) + acdd* (a+b)* = ∧* + ∅* + (a*b*)* (ab)*(c*+d*) = (ab)*(c+c*) + (ab)*( ∧ + d*)

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