Let K_n denote the simple graph on n vertices.
(a) Let v be some vertex of K_n and consider K _{n − v}, the graph obtained by deleting
v. Prove that K _{n − v} is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following statement:
K _n , the complete graph on n vertices, has n(n-1)/2
edges

Let W denote the set of English words. For u, v ∈ W, declare u ∼ v provided that u, v have the same length and u, v have the same first letter and u, v have the same last letter.

a) Prove that ∼ is an equivalence relation.

b) List all elements of the equivalence class [a]

c) List all elements of [ox]

d) List all elements of [are]

e) List all elements of [five]. Can you find more than 15?

f) Bonus. Find all three letter words x such that [x] has 5 elements.

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V₁ and V₂, such that every edge joins a vertex in V₁ with a vertex in V₂. This means no edges are within V₁ or V₂ (or symbolically: ∀u, v ∈ V₁, {u,v} ∉ E and ∀u,v ∈ V₂, {u,v} ∉ E).

8(a) Show that the complete graph K₂ is a bipartite graph.

8(b) Prove that no complete graph K_n, where n > 2, is a bipartite graph.

8(c) Prove that every rooted tree forms a bipartite graph.

1.-

A) A particle of total energy 9V₀ is incident from the -x-axis on a potential given by V(x) = 8V₀ for x < 0, V(x) = 0 for 0 < x < a and V(x) = 5V₀ for x > a. Find the probability that the particle will be transmitted on through to the positive side of the x-axis, x > a.

B) Consider a particle incident from the left on a potential step which is defined as V(x) = V₁ for x < 0 and V(x) = V₂ for x > 0 with V₁ < V₂. Find the solution of the Schrödinger equation for the following cases: (i) V₁ < E < V₂ (ii) E > V₂.

A metal sphere with radius ra is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius rb. There is charge +q on the inner sphere and charge −q on the outer spherical shell. Take V to be zero when r is infinite.

Calculate the potential V(r) for r<r a .

Calculate the potential V(r) for ra<r<rb

Calculate the potential V(r) for r>rb

Find the potential of the inner sphere with respect to the outer.

Use the equation Er=−∂V∂r and the result from part (b) to find the electric field at any point between the spheres (ra<r<rb).

Use the equation Er=−∂V∂r and the result from part (c) to find the electric field at a point outside the larger sphere at a distance r from the center, where r>

I will really appreciate it if you could answer all of them all for me. Thank you :)

1.

use a direct proof to SHOW the following:

The square of an even natural number is even.

The sum of an even and odd number is odd.

The sum of two even number is even.

The sum of two odd number is even.

2.

Examine below compound proposition:

[-p ^ ( p v q ) ] -> q.

(1) Complete truth table

(2) Explain if this IS or IS NOT a tautology, and why.

3.

Determine the satisfiability of the following compound proposition:

(p v -q) ^ (q v -r) ^ (r v -p) ^ (p v q v r) ^ (-p v -q v -r ).

4. Select ALL that is true about the following logical operation between 'p' and 'q.'

p ∧ q.

if, and only if, both p and q are T, the end result will be T (T/F)

only if both p and q are 1 will result in a 1 (T/F)

If only 1 of the variables is 1, the result CAN still be a 1 (T/F)

in 3 of the 4 outcomes, the outcome will be F. (T/F)

Here are four problems (5 pts each) involving the calculation of DG°' for metabolic reactions we have discussed, based on experimentally determined redox potentials. Calculate the DG°' for each reaction using the equation DG°' = -nFDE₀' and the values for E₀' given in Table 1. Show your work and circle your answer.

Table 1. Reduction potentials for reduction half reactions:

1/2 O₂ + 2H⁺ + 2e^- ® H₂O                                              E₀' = +0.82 V

fumarate + 2H⁺ + 2e^- ® succinate                                  E₀' = +0.03 V

oxaloacetate + 2H⁺ + 2e^- ® malate                                 E₀' = -0.17 V

pyruvate + 2H⁺ + 2e^- ® lactate                                       E₀' = -0.19 V

a-ketoglutarate + CO₂ + 2H⁺ + 2e^- ® isocitrate               E₀' = -0.38 V

FAD + 2H⁺ + 2e^- ® FADH₂                                             E₀' = -0.22 V

NAD⁺ + 2H⁺ + 2e^- ® NADH + H⁺                                    E₀' = -0.32 V

CoQ + 2H⁺ + 2e^- ® CoQH₂                                            E₀' = +0.06 V

Problem 1. isocitrate + NAD⁺ ® a-ketoglutarate + CO₂ + NADH

Problem 2. succinate + FAD ® fumarate + FADH₂

Here are four problems (5 pts each) involving the calculation of DG°' for metabolic reactions we have discussed, based on experimentally determined redox potentials. Calculate the DG°' for each reaction using the equation DG°' = -nFDE₀' and the values for E₀' given in Table 1. Show your work and circle your answer.

Table 1. Reduction potentials for reduction half reactions:

1/2 O₂ + 2H⁺ + 2e^- ® H₂O                                              E₀' = +0.82 V

fumarate + 2H⁺ + 2e^- ® succinate                                  E₀' = +0.03 V

oxaloacetate + 2H⁺ + 2e^- ® malate                                 E₀' = -0.17 V

pyruvate + 2H⁺ + 2e^- ® lactate                                       E₀' = -0.19 V

a-ketoglutarate + CO₂ + 2H⁺ + 2e^- ® isocitrate               E₀' = -0.38 V

FAD + 2H⁺ + 2e^- ® FADH₂                                             E₀' = -0.22 V

NAD⁺ + 2H⁺ + 2e^- ® NADH + H⁺                                    E₀' = -0.32 V

CoQ + 2H⁺ + 2e^- ® CoQH₂                                            E₀' = +0.06 V

Problem 3. malate + NAD⁺ ® oxaloacetate + NADH + H⁺

Problem 4. FADH₂ + CoQ ® FAD + CoQH₂