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...
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...
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
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...