Prove: If A is an uncountable set, then it has both uncountable and countably infinite subsets.

Please show that answer step by step and explain clearly, thx!!!!

5. Does the accuracy of a kNN classifier using the Euclidean distance change if you (a) translate the data

(b) scale the data (i.e., multiply the all the points by a constant), or (c) rotate the data? Explain.

Answer the same for a kNN classifier using Manhattan distance.

Answer all parts of question 1.

1a.) Find the limits, as t approaches both positive infinity and negative infinity, of the solution Φ(t) of the ivp x' = (x+2)(1-x^4), x(0) = 0

1b.) Find the value of a such that the existence and uniqueness theorem applies to the ivp x' = (3/2)((|x|)^(1/3)), x(0) = a.

1c.) Explain why x' + ((sin(t)) / ((e^t) + 1)) * x = 0 cannot have a solution x(t) such that x(1) = 1 and x(2) = -1.

3. Dictator and Ultimatum Games with Fehr-Schmidt Preferences. Now let the two utility functions be given by,

U_A = x_A − .6 max(x_B − x_A, 0) − β_A max(x_A − x_B, 0),

U_B = x_B − α_B max(x_A − x_B, 0).

(a) Suppose that the two agents play a dictator game in which player A is given an endowment of 100 and may transfer any amount s ∈ [0, 100] to player B. The material payoffs are then given by (x_A, x_B) = (100 − s, s). Compute player A's strategy for all possible values of β_A.

(b) Now suppose that the two agents are playing an ultimatum game with an endowment of 100. Write down the set of players, the pure strategy space of each player and the payoff functions.

(c) Calculate B's best response to every offer s ∈ [0, 100].

(d) How does the subgame perfect equilibrium of the game depend on α_B and β_A?

1. When solving for the reactions at the supports of a truss, what equations do you use?

2. When solving for the forces in a member of a truss, how do you know you assumed the sense (direction) of the force in the incorrect direction? You will get negetavie values

3. When using the method of joints and you are analyzing an individual joint, how many and which equations of equilibrium can you apply?

4. When using the method of sections, what is the maximum number of members you can cut?

5. When solving a friction problem, what is the equation used to find the force of friction?

§Radiant Transportation company provides two types of boats –Boat A and Boat B. Each Boat A can carry 15 passengers, 10 vehicles, and 40 tons of goods. On the other hand, Each Boat B can carry 15 passengers, 30 vehicles, and 30 tons of goods. You are planning to hire boats from Radiant as you have to ship 120 passengers, 120 vehicles, and 120 tons of Goods. If each Boat A costs $2000 and each Boat B costs $3000 then how many of each one you should hire to minimize cost while you ship the required number of passengers, vehicles, and goods.

§Keep in mind that you have to hire at least 1 of each type of boat.

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε).

(a) Show that every element of N∗(x;ε) is an interior point.
(b) Determine the boundary of N∗(x;ε) and prove your answer is correct.

1. Describe how to find the complement and supplement of an angle.
2. Explain how to find the measure of the third angle in a triangle if you know the measures of the other two angles.
3. Explain the Pythagorean Theorem is used to find the length of one side of a right triangle if the lengths of the other two sides is known.
4. Explain how you can use the shadow of a tall object to measure its height.

1) Use MATLAB to solve this differential equation. ??/?? = .25? (1 − ?/4 ) - a

2) Use MATLAB to graph solution curves to this system with several different initial values. Be sure to show at least one solution curve for each of the scenarios found in ??/?? = .25? (1 − ?/4 ) - a ( let a = 0.16)

solve using both methods (Dsolve and ODE45 on matlab) please provide steps

1) y'+y=e^x

2) y'+2y= 2 sin(x)

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R.

a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R).

ii). Show that the subset I of S = Mn(R) consisting of those matrices A = (aij ) ∈ Mn(R) with aij = 0 whenever 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1 is a left ideal of Mn(R).

b) Let R be a ring. An element a in R is called idempotent if a 2 = a.

i) Find all idempotent elements of the rings R = Z/6Z, and S = Z[i]

(ii) Show that if a is an idempotent in a ring R, then so is b = 1 − a.

(iii) Show that if R is a commutative ring, then the set of all idempotent elements of R is closed under multiplication

(iv) A ring B is a Boolean ring if a 2 = a for all a ∈ B, so that every element is idempotent. By considering (x + x) 2 show that 2a = 0 for any element a in a Boolean ring B.

v) Show that if B is a Boolean ring, then B is commutative.

vi) Show that if R is a commutative ring and a and b are idempotents, then a ⊕ b := a + b − ab 1 2 PROBLEM SHEET 4 is also an idempotent. Show that the set B = Idem(R) of all idempotents of R is a Boolian ring, where the addition is ⊕ and the multiplication is the same as in R.

vii) Let E be a set and let B the set of all subsets of E, show that B is a Booloian ring, where the ”multiplication” of two elements of B (i.e. subsets of E) is the intersection of these subsets, while the addition in B is given by X + Y = (X ∪ Y ) \ (X ∩ Y ) Here X, Y ∈ B (so, X ⊂ E, Y ⊂ E). What are the unit and zero elements of B?

if the involute is known show how to find the evolute ?