(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two...

(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two bounded subset of R. Define a set A − B := {a − b : a ∈ A and b ∈ B}. Show that sup(A − B) = sup A − inf B and inf(A − B) = inf A − sup B

Expert Solution

Colby Messinger answered 2 years ago

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes....

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

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 V be the set of positive reals, V = {x ∈ R : x >...

Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

Let R=R+. Define: a+b = ab ; ab = a^(lnb) 1. Is (R+, +, ) a...

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Let V = { S, A, B, a, b, λ} and T = { a, b...

Let V = { S, A, B, a, b, λ} and T = { a, b }, Find the languages generated by the grammar G = ( V, T, S, P } when the set of productions consists of: S → AB, A → aba, B → bab. S → AB, S → bA, A → bb, B → aa. S → AB, S → AA, A → Ab, A → a, B → b. S → A, S →...

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

6. Let V be the vector space above. Consider the maps T : V → V...

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1. Let V and W be vector spaces over R. a) Show that if T: V...

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(10pt) Let V and W be a vector space over R. Show that V × W...

(10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W and λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv. (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information...

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a...

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