For p, q ∈ S^1, the unit circle in the plane, let d_a(p, q) = min{|angle(p)...

For p, q ∈ S^1, the unit circle in the plane, let
d_a(p, q) = min{|angle(p) − angle(q)| , 2π − |angle(p) − angle(q)|}
where angle(z) ∈ [0, 2π) refers to the angle that z makes with the positive x-axis.
Use your geometric talent to prove that d_a is a metric on S^1.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Prove p → (q ∨ r), q → s, r → s ⊢ p → s

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, ....

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, . . . , [p-1]p}. Prove that for y ≠ 0, Ly restricts to a bijective map Ly|s : S → S. 11.5 Prove Fermat's Little Theorem

Let C be a plane curve parameterized by arc length by α(s), T(s) its unit tangent...

Let C be a plane curve parameterized by arc length by α(s), T(s) its unit tangent vector and N(s) be its unit normal vector. Show d dsN(s) = −κ(s)T(s).

Let p and q be propositions. (i) Show (p →q) ≡ (p ∧ ¬q) →F (ii.)...

Let p and q be propositions. (i) Show (p →q) ≡ (p ∧ ¬q) →F (ii.) Why does this equivalency allow us to use the proof by contradiction technique?

Consider a circle with AB as diameter, and P another point on the circle. Let M...

Consider a circle with AB as diameter, and P another point on the circle. Let M be the foot of the perpendicular from P to AB. Draw the circles which have AM and MB respectively as diameters, which meet AP at Q and BP at R. Prove that QR is a tangent to both circles.

Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t}...

Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t} and B = {r, s, t, u} are events. x p q r s t u p(x) 0.15 0.25 0.2 0.15 0.1 (a) Determine what must be p(s). (b) Find p(A), p(B) and p(A∩B). (c) Determine whether A and B are independent. Explain. (d) Arer A and B mutually exclusive? Explain. (e) Does this table represent a probability istribution of any random variable? Explain.

Let p and q be two linearly independent vectors in R^n such that ||p||_2=1, ||q||_2=1 ....

Let p and q be two linearly independent vectors in R^n such that ||p||_2=1, ||q||_2=1 . Let A=pq^T+qp^T. determine the kernel, nullspace,rank and eigenvalue decomposition of A in terms of p and q.

Let D = demand, S = supply, P = equilibrium price, and Q = equilibrium quantity....

Let D = demand, S = supply, P = equilibrium price, and Q = equilibrium quantity. What happens in the market for electric vehicles if the government offers incentives to manufacturers to produce more electric vehicles? Provide a graphical representation to your answer in question 5. The graph can either be hand-drawn or copied from the textbook or other online sources.

Market demand (D) and supply (S) are:D: P = 60-2QandS: P = 5 +Q. Let Qe...

Market demand (D) and supply (S) are:D: P = 60-2QandS: P = 5 +Q. Let Qe = Quantity at equilibrium and Pe = Priceat equilibrium.(a) Compute Qe and Pe and graph the D and S functions in the same graph with P on the vertical axis. (b) Show that at Q1 = 9, Net Market Benefits (NB) are less than NB at Qe. What is the Dead-Weight Loss (DWE) at Q1 = 9? (c)Show that at Q2 = 20, Net...

FOR EAICH PAIR OF PROPOSITIONS P AND Q STATE WHETHER ON NOT p=q p=(s→(p ∧¬r)) ∧...

FOR EAICH PAIR OF PROPOSITIONS P AND Q STATE WHETHER ON NOT p=q p=(s→(p ∧¬r)) ∧ ((p→(r ∨ q)) ∧ s), Q=p ∨ t

Subjects