Homework Statement For any integer n, let A(n) be the statement: “If n 2 = 4k + 1 for some k ∈ Z, then n = 4q + 1 or 4q + 3 for some q ∈ Z.” Use proof by contradiction to show that A(n) is true for all n ∈ Z. The Attempt at a Solution [/B] the answer sheet says that since n !=4q+1 and n !=...
2. ### B My First Post: A Contradiction in Simple Free Fall?

Hello, Let us imagine a solid in a free fall on planet Earth and let us neglect the effect of air resistance on the motion of the solid. According to Newton's Second Law of Motion, since the only force being applied on the solid is its Weight, then its acceleration must be its Weight divided by...
3. ### I Different answers from equation of continuity and Bernoulli

Hi there, So I was doing the dishes this morning using a sink wand hat can toggle between different flow speeds. The way that I've always thought of this working is using the equation of continuity: Volume flow rate: = Area*velocity Pressing a button on the wand decreases the cross-sectional...
4. ### Force exerted on the horizontal bottom of the container

Homework Statement In an empty container, having the shape of a Rectangular Parallelepiped, with width l=0.5m and MN=H=0.4m, and NP=L=0.75M. we poureda mass of mercury =2448kg. T0,T1,T2 have the same section which is 0.2 m^2. we give g=10N/kg, and density of mercury u=13.6 g/(cm^3)...
5. ### I Ions passing through a parallel plate capacitor with a varying voltage

Consider the scenario Two parallel plates One ion particle (mass 1, charge +1) traveling parallel to the plates through the middle. Voltage of the plates is the same. As the ion travels between the plates the voltage on the plates ramps up, and the electric field between the plates remains...
6. ### I Thought Experiment Contradiction

Hello all, I have a question that's been bothering me the last few days and wasn't sure where to turn. Recall the original Special Relativity thought experiment: A spaceship travels at constant velocity v, moving in the positive x direction. An observer on the spaceship emits a photon directly...
7. ### I Proof by contradiction

Hello. I'm currently studying about natural numbers and I encountered the theorem of definition by recursion: This states: Let ##H## be a set, let ##e \in H## and let ##k: H \rightarrow H## be a function. Then there is a unique function ##f: \mathbb{N} \rightarrow H## such that ##f(1) = e##...
8. ### I E cross B field, with E>B

I have been unable to find a satisfactory explanation of this problem, elsewhere. Consider an uniform electric field, E, along the y axis. Consider also a uniform magnetic field, B, along the z axis. If we release a particle (charge=q, mass=m) at rest on the origin at time t=0, what will be...
9. ### Bernoulli's Equation & Equation of Continuity

Hi all, I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity. What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase...
10. ### Transformation equation of velocity contradicts stellar aberration?

Let S be the frame where the Sun is at rest. Imagine light from the North Star reaches the centre of the Sun, and let's define the equatorial plane as the plane that is perpendicular to this light and cuts the Sun into two hemisphere. Suppose a distant star A is on this equatorial plane and its...
11. ### Composition of 2 transformations of E

Suppose frame ##S^\prime## moves in the positive ##x## direction at ##v## with respect to frame ##S##, and frame ##S^"## moves in the positive ##y## direction at ##v^\prime## with respect to frame ##S^\prime##. Then, ##E^\prime_x=E_x## ##E^\prime_y=\gamma(E_y - vB_z)## ##E^\prime_z=\gamma(E_z...
12. ### Prove that if a < 1/a < b < 1/b then a < -1.

Hello again. I recently submitted a thread asking for feedback on a couple of very basic proofs for an exercise from the book "How To Prove It" by Velleman. This is another request for you to help me understand how wrong my proof for a new exercise could be improved. 1. Homework Statement...
13. ### Proofs involving negations and conditionals

0. Background First and foremost, this is a proof-reading request. I'm going through Velleman's "How To Prove It" because I found that writing and understanding proofs is a prerequisite to serious study of mathematics that I did not meet. Unfortunately, the book is very light on answers to its...
14. ### "Effect of couple is ind. of point of appl."; then how do we explain the following CONTRADICTION?

We read that the a couple is a free vector, i.e it does not depend on the point of application. However, please consider the following example: If you apply two equal magnitude forces anti-parallel to each other, one on each side of the wheel, a couple is formed, and so the wheel will rotate...