Record some Problems

In this post I want to record 3 problems:

The following is a problem in GRE math subject test:

Problem 1. Let $\F_p$ be a finite field of order $p$, show that the number of noninvertible $2\times 2$ matrices over $\F_p$ is \[p^4-p^3-p^2+p.\]

Can we generalize to $n\times n$ matrices?

The following is a problem found somewhere by someone in Taiwan's forum, actually I also find this in mainland textbook on real functions:

Problem 2. Let $E,E_1,E_2\subseteq \R$ be such that $E=E_1\cup E_2$. If $E$ is measurable such that \[
m(E)=m^*(E_1)+m^*(E_2)<\infty,
\] show that $E_1$ and $E_2$ are measurable.
HINT: Approximate $E_1,E_2$ from outside by measurable set.

The following is a problem in the midterm exam of Math4061 (fall 2013), which I want to record a simple proof here.

Problem 3. Let $X,Y$ be metric spaces. Suppose that $f:X\to Y$ is a function such that

• $f$ is 1-1.
• $K$ compact in $X$ $\implies$ $f(k)$ compact in $Y$.

Prove that $f$ is continuous.

Dr Li's original solution proves by contradiction, I want to give a direct proof using the sequential criterion of continuity for metric spaces.

Solution to Problem 3. Suppose that $x_n\to x$ in $X$, let's prove that $f(x_n)\to f(x)$ in $Y$. First we observe that \[
L:=\{x_n:n\ge 1\}\cup \{x\}
\] is compact. Second we define $g:=f|_{L}:L\to f(L)$ and show that $g^{-1}:f(L)\to L$ is continuous. Indeed, let $K\subseteq L$ be any closed set, since $L$ is compact, so is $K$, and thus \[
f(K)=g(K)=(g^{-1})^{-1}(K)
\] is a compact set, hence closed set, in $Y$ by hypothesis. Therefore any closed set in $L$ is pulled back to a closed set in $f(L)$, $g^{-1}$ is continuous. Since $f(L)$ is compact and $L$ is Hausdorff (as it is a metric space), a standard training in topology enables us to show $g^{-1}$ is a homeomorphism, so $g$ is continuous.

Finally we check that $f(x_n)\to f(x)$. Since $x_n\to x$ in $X$, we have $x_n\to x$ in $L$, so $g(x_n)\to g(x)$ in $f(L)$, but then $f(x_n)=g(x_n)\to g(x)=f(x)$ in $Y$, so we are done.$\qed$