1. ### Here's an extension of a list posted earlier. If anybody can think of

oh yay, I'm glad people like this :).
2. ### Rational numbers and Lowest terms proof:

You're on the right track. Let r = p_1 * p_2 * ... * p_a. Let s = q_1 * q_2 * ... * q_b. Let t = p'_1 * p'_2 * ... * p'_c. Let u = q'_1 * q'_2 * ... * q'_d. Cross multiply to get (p_1 * p_2 * ... * p_k)(q'_1 * q'_2 * ... * q'_k) = (p'_1 * p'_2 * ... * p'_k)(q_1 * q_2 * ... *...
3. ### Difference hard to notice ?

The word 'homogeneous' entails that all constant terms are zero, which is not required for an algebraic equation.
4. ### Testing a series for convergence/divergence

0 \leq \frac{2+(-1)^n}{n^2+7} \leq \frac{3}{n^2+7}. You can squeeze it as such. If the series on the right converges, then so does yours.
5. ### Here's an extension of a list posted earlier. If anybody can think of

Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D! Perspectives of the world: ------------------------------- Optimist – The glass is half-full. Pessimist – The glass is half-empty. Existentialist – The glass is. Fatalist – The...
6. ### Science Jokes

Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D! Perspectives of the world: ------------------------------- Optimist – The glass is half-full. Pessimist – The glass is half-empty. Existentialist – The glass is. Fatalist – The...
7. ### Cylindrical coordinates to cartesian coordinates

The equation for the plane is just y = \sqrt{3}x.
8. ### Abstract Math, Tautology

Note: ~(P v Q) is equivalent to (~P and ~Q). So your sample statement S does not work.
9. ### Please explain the solution (Real Analysis)

Do you understand why \textbf{N}^k is countable?
10. ### Volume of a rectangle by cross-sections

In the pyramid integration, x varies from 0 to 3. However, each cross section of a rectangular prism has the same base. So you would have instead: \int_0^3 (3)^2 dx
11. ### Define the sigma-algebra generated by a partition

Yes, that should work.
12. ### Volume of a rectangle by cross-sections

First of all, there is no such thing as the "volume of a rectangle." If you mean a rectangular prism, then the volume will be different. A rectangular prism with the same square base and height would have a volume of 27 cubic units. V_{rectangular \ prism} = bh. V_{pyramid} = \frac{bh}{3}...
13. ### How to increase max speed of wave

I believe we would want (e) and not (f). A thicker string would result in a greater affinity for the string to return back to its equilibrium position. Think of it as: It takes more effort to pull a thicker string, so it'll have a 'stronger desire' to return back to its original position.
14. ### Exhibit a bijection between N and the set of all odd integers greater than 13

Almost...your injection proof is fine. But, when proving a mapping is surjective, we need to show that the domain maps all of the range, i.e., show that for each y, there's an x such that f(x) = y. You wrote "for y E N," but that's not true. y E B = set of all odd integers greater than 13, not...
15. ### Laplace transform of phase-shifted sinusoid

Use trig summation identity to simplify: f(t) = \left[ sin(4t)cos(\frac{\pi}{3}) + cos(4t)sin(\frac{\pi}{3}) \right] u(t). Can you finish from here?