What is Equipotent relation?
In Royden's book Real Analysis, page 13, he writes that "We call two sets A and b equipotent provided there is a one-to-one mapping $f$ from A onto B...
Category: General
Calculus question with circle, and string tracing an area
The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Fi...
How to plot a delta function in Matlab
How to plot this function in Matlab: $S(f)=1+\frac{1}{4}(\delta(f+2f{_1})+\delta(f-2f{_1})-\delta(f+2f{_2})+\delta(f-2f{_2}))$ Thanks for any advice!
In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?
Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&...
Is empty set element of every set if it is subset of every set?
This problem is from Discrete Mathematics and its Applications My question is on 9b. I know that the sign represents an element is a member of. (from book...
Understanding the definition of a splitting field
The splitting field of a polynomial $p$ with coefficients over a field $K$ is defined as the smallest field that contains $K$, in which the polynomial can...
Using unit circle to explain $\cos(0) = 1$ and $\sin(90) = 1$
We have been taught $\cos(0) = 1$ and $\sin(90) = 1$. But, how do I visualize these angles on the unit circle?
Understanding how to prove when a subset is a subgroup
Lemma 3.4. Let $(G ,*)$ be a group. A nonempty subset $H$ of $G$ is a subgroup of $(G,*)$, iff, for every $a, b\in H$, $a*b^{-1}\in H$. Proof. First, supp...
Partial Differential Equation with a flux term
Recall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm dt}\int_a^b(\text{quantity) d}x=\big[\text{Flux...
Definition of continuous map
Let $X,Y$ be topological spaces, let $f:X\longrightarrow Y$ a function. There are several (equivalent) ways to define continuity of $f$, one of these says...
