Suppose K is a nonempty compact subset of a metric space X and x ∈ X.

(i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x).

(ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x).

[Suggestion: As a start, let S = {d(x, y) : y ∈ K} and show there is a sequence (qn) from K such that the numerical sequence (d(x, qn)) converges to inf(S).] 63

(iii) Let X = R \ {0} and K = (0, 1]. Is there a point x ∈ X with no closest point in K? Is K closed, bounded, complete, compact?

(iv) Let E = {e0, e1, . . . } be a countable set. Define a metric d on E by d(ej , ek) = 1 for j not equal k and j, k not equal 0; d(ej , ej ) = 0 and d(e0, ej ) = 1 + 1/j for j not equal 0. Show d is a metric on E. Let K = {e1, e2, . . . } and x = e0. Is there a closest point in K to x? Is K closed, bounded, complete, compact?

3. For each of the following relations on the set Z of integers, determine if it is reflexive, symmetric, antisymmetric, or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

(a) R = {(a, b) ∈ Z 2 ∶ a 2 = b 2 }.

(b) S = {(a, b) ∈ Z 2 ∶ ∣a − b∣ ≤ 1}.

proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)

Consider the group homomorphism φ : S₃ × S₅→ S₅ and φ((σ, τ )) = τ .

(a) Determine the kernel of φ. Prove your answer. Call K the kernel.

(b) What are all the left cosets of K in S₃× S₅ using set builder notation.

(c) What are all the right cosets of K in S₃ × S₅ using set builder notation.

(d) What is the preimage of an element σ ∈ S₅ under φ?

(e) Compare your answers in parts (b)-(d). What do you notice?

An element e of a ring is called an idempotent if e^2 = e. Find a nontrivial idempotent e in the ring Z143.

Problem 2. Consider a graph G = (V,E) where |V|=n.

2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates.

2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t?

2(c) What is the total number of possible cycles of any length in G from a given starting vertex s?

Solve the following wave equation using Fourier Series

a²u_xx = u_tt, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx, u_t(x,0) = pi - x

Solve the following wave equation using Fourier Series

a²u_xx = u_tt, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x², u_t(x,0) = 0

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < 1, t > 0, u_x(0,t) = 0 = u_x(1,t), u(x,0) = 1 - x²

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G.

iii. What is the total degree of a tree?

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx - sin3x

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x/2