Topics In Applied Math

Rudin Principles 1.15.a. Prove: $x \not = 0 \land xy = xz \implies y=z$ $$ \begin{array}{l c} y =y & \\ y = y \times 1 & multiplicative \ identity\\ y = y \times (x \times x^{-1}) & multiplicative \ inverse \ x \times x^{-1} = 1 \\ y = (y \times x) \times x^{-1} & associativity \\ y = (z \times x) \times x^{-1} & substitute \ given \ xy=xz \\ y = z \times (x \times x^{-1}) & associative \ given \ xy=xz \\ y = z \times 1 & multiplicative \ inverse \\ y = z & multiplicative \ identity \end{array} $$ 1.15.b. Prove: $x \not = 0 \land xy = x \implies y=1$ $$ \begin{array}{l c} y = y & \\ y = y \times 1 & multiplicative \ identity\\ y = y \times (x \times x^{-1}) & multiplicative \ inverse \ x \times x^{-1} = 1 \\ y = (y \times x) \times x^{-1} & associativity \\ y = x \times x^{-1} & substitute \ given \ xy=x \\ y = 1 & multiplicative \ inverse \end{array} $$ 1.15.c. Prove: $x \not = 0 \land xy = 1 \imp...

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=...

Walter Rudin's principles of Mathematical Analysis is generally considered the de facto standard textbook for the subject. But it is not an easy read. As I am going through several chapters of it, I thought it may be of use to others to post some further details on the material. I am using his third edition textbook. Professor Rudin's style is terse. Although the result is concise and elegant, detailed steps are usually omitted from his proofs. So I plan to post some of those extra details here along with other relevant information. Besides any supplementary info on Rudin's book, I also plan to post on other topics in analysis.