Guide To Walter Rudin's Principles, 1.14 (Proof Details)

Proofs - Rudin Principles 1.14

It is useful to read about Abelian groups first. In particular, review inverse and identity elements. Any book on modern or abstract algebra should cover this. In the field of reals $\mathbb{R}$, prove:

1.14.a. if $ x+y = x+z$ then $y=z$
We need to prove $y=z$ under the condition given. Rudin starts with $y=y$ (although not shown in the text) $$ \begin{array}{l c} y = y & \\ y = 0+y & additive \ \ identity\\ y = (x-x) +y & additive \ \ inverse\\ y = x - (x+y) & associative \ \ property\\ y = x - (x+z) & equation \ \ given \ \ (substitute \ \ x+y = x+z )\\ y = (x - x)+z & associative \\ y = 0 + z & additive \ \ inverse \\ y=z \end{array} $$ 1.14.b. if $ x+y = x$ then $y=0$
Note: this proof asserts uniqueness of the additive identity element.
Why can we not substitute $y=0$ right away? We would get $x+0=x$ and prove $x=x$.
Rudin says substitute $z=0$ in 1.14.a and follow the same path. So starting with $y=y$ $$ \begin{array}{l c} y = y & \\ y = y + 0 & additive \ \ identity\\ y = y + (x - x) & \\ y = (y + x) - x & \\ y = x-x & substitute \ \ x+y = x\\ y=0 \end{array} $$ 1.14.c. if $ x+y = 0$ then $y=-x$
$$ \begin{array}{l c} y = y & \\ y = y + 0 & additive \ \ identity\\ y = y + (x - x) & \\ y = 0 -x & substitute \ \ x+y = 0\\ y = -x & \end{array} $$ 1.14.d: prove $-(-x)=x$
Rudin says start with (1.14.c) and put $-x$ where $x$ is:
if $ -x+y = 0$ then $y=-(-x)$ but we interpret this as ($y=x$)
if $-x + x = 0 $ then $x = -(-x)$
Additive inverse definition is $x + (-x) = 0 $ and putting $-x$ for each $x$ we get $-x + (-(-x)) = 0 $
$$ \begin{array}{l c} x = x & \\ x = x + 0 & additive \ identity \ x + 0 = x\\ x = x + (-x) - (-x) & additive \ inverse \ (-x) + (- (-x)) = 0\\ x = (x + (-x)) - (-x) & associativity \\ x = 0 - (-x) & additive \ inverse \ x + (-x) = 0 \\ x = -(-x) \end{array} $$

Comments

Post a Comment

Popular posts from this blog

Data science & ML Video Tutorials Part II - Group & Set Theory (Groups, Rings & Fields)

Image
We continue building the knowledge required for basic probability theory and statistics (needed for data science & ML). The videos in this post are about set algebra and algebraic structures like groups, rings & fields. It includes an introduction to identity, inverse and idempotent elements. We'll see that Boolean conjunction and disjunction can be seen as multiplication and addition and investigate possible additive and multiplicative identities and inverses. This is followed by power sets and basic operations on elements of power sets (union, intersection, complement, difference). All concepts are then combined in a discussion of algebraic structures including groups, rings and fields. We end with Boolean rings and describe set algebra as an example Boolean ring. In the review session we complete an example proof from CH1 of W. Rudin's Principles Of Mathematical Analysis using the axioms of a field. Introduction to part II. The main subject of the sessions of Par
Read more

Finite Difference Simulation of a Helmholtz Resonator - 2D Acoustic Wave Equation

In this project we wish to simulate the acoustics of a Helmholtz Resonator (HR) as may be found in a vented loudspeaker enclosure design. The vent functions as a Helmholtz resonator and has a low frequency attenuation effect. The enclosure is compartmentalized due to the vent design. Therefore, in order to carry out the simulation we must implement reflecting boundary conditions in the interior of the simulation region. In addition, a mixture of absorbing and reflecting boundary conditions are needed at the edges of the simulation region. Absorbing boundary conditions for the wave equation require additional attention and are implemented as described by [Alam & Mohiuddin, 2021]. Read a detailed explanation report can be found here here . The Github project is availabe here: https://github.com/aruymgaart/AMATH/tree/master/acoustic_simulation .
Read more

Combination Finite Difference & Spectral Solution to Advection-Diffusion PDE

Vorticity $\omega$ is a measure of local rotation (in a fluid). It is related to the angular velocity of a solid. Vorticity is generally a vector (valued function). However, it can be viewed as a scalar in 2D because it is always perpendicular to the plane. If fluid velocity $\vec{v}$ (in 2D) has components $u$ and $v$ \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} \end{equation} then these vector components are related to a scalar quantity called the \textbf{stream function} $\psi$ as follows \begin{align} u = -\psi_y \\ v = \psi_x \end{align} so that \begin{equation} \vec{v}(t) = \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = \begin{pmatrix} -\psi_y \\ \psi_x \end{pmatrix} \end{equation} So $\psi$ is a scalar valued function while the components of the fluid velocity vector can be derived from it. \begin{equation} \label{eq:hw4_laplacian_eqn} \nabla^2 \psi = \omega \end{equation} Vorticity-Streamfunction PDE \begin{align} \omega_t + [\psi,\omega] &am
Read more