Recent content by The1TL

Homework Statement is there a continuous real valued variable X with mgf: (1/2)(1+e^t) Homework Equations The Attempt at a Solution I've noticed that this is the mgf of a bernoulli distribution with p =1/2. But since bernoulli is a discrete distribution, does that disprove that...

actually, I've noticed that the function g(x) = sqrt(x^2 + y^2) has the property that -y(df/dx) + x(df/dy) = 0. Therefore I could let F be h(g(x)) where h is the identity function.

oh ok, so for b) is there some way that I could show that F exists based on the -y(df/dx) + x(df/dy) = 0 equation? Or would it simply involve turning sqrt(x^2 + y^2) into an R^2 vector?

Homework Statement Suppose f:R^2 - {0} → R is a differentiable function whose gradient is nowhere 0 and that satisfies -y(df/dx) + x(df/dy) = 0 everywhere. a) find the level curves of f b) Show that there is a differentiable function F defined on the set of positive real numbers so that...

Homework Statement Suppose f: R^2 --> R is differentiable and (df/dt) = c(df/dx) for some nonzero constant c. Prove that f(x, t) = h(x + ct) for some function h. Homework Equations hint: use (u, v) = (x, x+ct) The Attempt at a Solution df/dt = limk-->0 (f(x, x+ct+k) - f(x...

Homework Statement Suppose f: R -> R is differentiable and let h(x,y) = f(√(x^2 + y^2)) for x ≠ 0. Letting r = √(x^2 + y^2), show that: x(dh/dx) + y(dh/dy) = rf'(r) Homework Equations The Attempt at a Solution I have begun by showing that rf'(r) = sqrt(x^2 + y^2) *...

Homework Statement Show that given some ε > 0, there exists a natural number M such that for all n ≥ M, (a^n)/n! < ε Homework Equations The Attempt at a Solution Ok so I know this seems similar to a Cauchy sequence problem but its not quite the same. So I am looking for a...

Does anybody know if I am correct? I'm not sure if I'm skipping steps.

Homework Statement Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.Homework Equations The Attempt at a Solution Let...

please help I've been trying for so long

Homework Statement Let f : A → B, g : B → C be functions where A,B,C are sets. ConsiderΓf ⊂A×B,the graph of f,Γg ⊂B×C,the graph of g. Now consider the sets Γ f ×C ⊂ A×B×C and A×Γg ⊂A×B×C. LetΓ=θ(Γf ×C∩A×Γg)⊂A×C where θ : A×B×C → A×C is the projection defined as θ((a,b,c))=(a,c). Show that Γ...

Homework Statement If f ∈ C(R) with f(0) ≠ 0, show that there exisits a g ∈ C(R) such that [fg] = [1], where [1] denotes the equivalence class containing the constant function 1. Homework Equations The Attempt at a Solution Let f ∈ C(R) such that f:R → R is defined as f(x) = 1/x...

Homework Statement prove that if a~a' then a+b ~ a' + b Homework Equations The Attempt at a Solution I can prove that if a=a' then a+b = a' + b but how can I apply this to any equivalence relation

Homework Statement Define a relation on R as follows. For a,b ∈ R, a ∼ b if a−b ∈ Z. Prove that this is an equivalence relation. Can you identify the set of equivalence classes with the unit circle in a natural way? Homework Equations The Attempt at a Solution I have already proven that this...

Homework Statement Let S1 = {a} be a set consisting of just one element and let S2 = {b, c} be a set consisting of two elements. Show that S1 × Z is bijective to S2 × Z. Homework Equations The Attempt at a Solution So I usually prove bijectivity by showing that two sets are...