Recent content by A.Magnus

I am working on a classical real analysis problem as follow: The answers from solution manual are respectively $ int (A) = \emptyset$ and $bd (A) = \{0\} \cup \{ \frac{1}{n} | n \in \mathbb N \}$. And here are my textbook's definition of interior point and border point: And then there is...

How about these revised steps - hope these come out right: (a) Since $x \neq 0$, there must exist $\frac{1}{x}$ such that $x \cdot \frac{1}{x} = 1$. (Multiplicative Inverse Property) (b) By Commutative Property, we have $\frac{1}{x} \cdot x = 1$. (c) Multiplying both sides of the equation by...

Thank you for your response! ~MA

Thank you for your helps! ~MA

I am working on a proof problem on ordered field from a textbook, which lists additive and multiplicative properties similar to the ones here: The followings are what I was able to come out -- I just wanted to make sure that they are acceptable: (a) By the multiplicative inverse property...

I am reading a chapter section on Ordered Field that starts off with the additive and multiplicative properties: To my untrained eyes, they do not mean anything at all. Could somebody therefore give an intuitive significance of the two properties, perhaps with examples - please. Are they about...

Thank you! Phew! Finally I got one proof right. ~ MA

The problem is actually a theorem that the textbook assigns as exercise. It is under the chapter section titled "The Ordering of Cardinality." I believe you are right, the paragraphs above this theorem make lots of reference that both $A, B$ are finite sets. Thank you. Just to recap what you...

I am working on a proof problem and I would love to know if my proof goes through: If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$. Proof: (a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not...

Opps! I don't understand why my mind strayed off. I am sorry. Let's do these: (a) If $x \in S \backslash T$, the $f: (S \backslash T) \rightarrow (T \backslash S)$ is given as bijection by the problem. (b) If $x \in S \cap T$ on the other hand, the $F(x) = x$ is an identity function and...

Thank you again for pointing this function out. The easiest way I can think of proving $F$ is bijective is by showing that both $F$ and $F^{-1}$ is injective. Is there any other simpler way? Thank you again for your time and gracious helps. ~MA

Thank you for your time. Apparently my logic was flawed. ~MA

I am working on a set equivalent (the textbook refers as "equinumerous" denoted by ~) as follows: If $S$ and $T$ are sets, prove that if $(S\backslash T) \sim (T\backslash S)$, then $S \sim T$. Here is my own proof, I am posting it here wanting to know if it is valid. (It may not be as elegant...

I am reading Larson's Calculus textbook and come across this paragraph about vector-valued function: The parametrization of the curve represented by the vector-valued function $$\textbf{r}(t) = f(t)\textbf{i} + g(t)\textbf{j} + h(t)k$$ is smooth on an open interval $I$ when $f'$, $g'$ and...

Thank you for your gracious helping hand, apologize for getting back to you late. ~MA