There are five distinct white and seven distinct blue shirts in a wardrobe. Find the number of ways of taking four shirts from the wardrobe such that a) they could be either white or blue, b) they are all white, c) they are all blue, d) they are all the same color and e) 2 are white and 2 are blue.

a. Use mathematical induction to prove that for any positive integer ?, 3 divide ?^3 + 2?
(leaving no remainder).
Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b + 3??^2 + ?^3.
b. Use strong induction to prove that any positive integer ? (? ≥ 2) can be written as a
product of primes.

Is it true that every symmetric positive definite matrix is necessarily nonsingular? (Need to show some form of proof, not just yes or no answer)

Please add some commentary for better understanding.

The Fellowship of the Ring traveling through the mountains of Moria (which are described by the functionf(x,y)=12x2y2e(-x2-y2)) and they find themselves at the point A=(1.5, -1, 1.05). They need to get to the top of the mountain they are climbing so they decide to travel along the plane y=-1. How steep is their climb when they start at point A? A diagram: https://www.geogebra.org/3d/jbkkyugx

1. What if they decided to go in a different direction, say they started walking towards the point (0,0,0) on a plane that is “vertical”. How steep would be the mountain going in that direction?

2. What is the general equation for planes that are vertical, like the one that you used to travel from A to (0,0,0)?

3. What would change in your calculations if we change where point A is? Can you generalize how to find the steepness in different directions from a given point A?

Let p be an integer other than 0, ±1.

(a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1.

(b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

Given the vector function r(t)=〈√t , 1/(t-1) ,e^2t 〉 a) Find: ∫ r(t)dt b) Calculate the definite integral of r(t) for 2 ≤ t ≤ 3

can you please provide a Matlab code?

Determine whether it is true of false and justify your answer.

Every nonempty bounded set S of real numbers has a supremum and infimum, but those might not be elements of the set.

Topic: Differential Equations; reduction order.

The indicated fuction y₁(x) is a solution of the given differential equation. Use the reduction order technique to find a second solution y₂(x).

xy'' + y' = 0;    y₁= ln(x)

Please, explain deeply each part of the solution, and include complete algebraic explanation as well. Otherwise, the answer will be negatively replied.

Propose two of your own random number generation schemes. Please generate 100 random numbers ?? (? = 1,2, … ,100) for each scheme and show the results on the same plot for comparison (i.e., x-axis of the plot will show the index ? and y-axis will show the generated random numbers ??. You can use different colors and/or symbols to distinguish one sequence from the other). Discuss which scheme will be preferred.

MatLab code for Bisection and Newton Method to find root for f(k) = p(k)^3+2*(p(k)^2)-10 on interval [-1,2]. P(o)=1.5, P(1)=2.

Answer=
1.654249

Some species of rodents live on average 6 years. In the first two years of life they do not reproduce and only 20% of the females born alive reach 2 years of age. Between the third and fourth year females have 1 daughter on average and their survival is 50%, while in the las two years they have 2 daughters on average. If in an ecosystem there is initially a population of 10 females between 0 and 2 years old, 4 between 3 and 4 years and 3 with more than 4 years. After 3 years the estimated population of females will be:

Instructions: When calculating, round the result to the nearest whole number.

- females between 0 and 2 years:_____

- females between 2 and 4 years:_____

- females with more than 4 years: ____

Consider the formula

A : ∃x.[(∀y.P(x, y) → R(x)) → ¬∃z.Q(x, z)]

(a) Find a formula equivalent to A that only has negation symbols in front of basic formulas.

(b) Give an example of an interpretation where A is true. The domain should be the set N.

(c) Give an example of an interpretation where A is false. The domain should be the set N.

fx to fx-k define wrt graph shift and show it for 1/x to 1/x-1 along with domain and range

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Construct the truth-table for the following propositional formulas. In each case, explain whether the formula is a tautology, a contradiction, or neither. (Explain how you arrive at this conclusion.) (a) ¬((p → ¬p) → ¬p) (b) (p → (q ∧ r)) → (¬r → ¬p) (c) (p → ¬q) → ¬(¬p → q)

She uses e-commerce or she pays by credit card. She does not pay by credit card. Using a truth table we get result ending T T F F with the T matching with T's in premsis so it is valid,

but in this problem the only Ts from the premises match with the T in ending column of truth table MY QUESTION IS WHY IS THIS INVALID WHILE THE ABOVE IS VALID DOES IT HAVE TO BE A TATUOLOGY TO BE VALID OR MATCH WITH Ts FROM PREMESIS

If you use your seat belt, you will be safer. You don’t use your seat belt. Therefore you won’t be safer.