23666: HW 7
Problem 30. Consider the lexicographical order on $\mathbb{R}\times\mathbb{R}$: $(x,y)\preceq (x',y')$ if $x\leq x'$ or ($x=x'$ and $y\leq y'$). Does every subset of $\mathbb{R}\times\mathbb{R}$ that is bounded from above with respect to this order have a least upper bound? Give a proof or provide a counterexample!
Problem 31. Let $\underline{a}<\overline{a}$ and $\underline{b}<\overline{b}$ be real numbers, and let $R$ denote the rectangle \[R=\{(x,y)\in\mathbb{R}^2\ |\ \underline{a}\leq x\leq \overline{a}, \ \underline{b}\leq y\leq \overline{b}\}.\] Denote by $\mathcal{H}$ the collection of all rectangles $S$ such that (1) $S$ is contained in $R$, (2) $S$ has sides parallel to the coordinate axes, and (3) $S$ has positive area.
Show that every non-empty subset $\mathcal{K}$ of $\mathcal{H}$ has a least upper bound.
Problem 32. Let $X$ be a fixed non-empty set. A function $f:X\to\mathbb{R}$ is called a characteristic function if there is a subset $A\subseteq X$ such that $f(x)=1$ if $x\in A$, and $f(x)=0$ if $x\not\in A$. In this case we write $f=\chi_A$.
- Show that the product of two characteristic functions is a characteristic function.
- Show that the sum $\chi_A+\chi_B$ of two characteristic functions is a characteristic function if and only if $A\cap B=\emptyset$. In this case $\chi_A+\chi_B=\chi_{A\cup B}$.
Problem 33.
- Find a function whose domain is the set of real numbers $\mathbb{R}$ and whose range is the set of rational numbers $\mathbb{Q}$.
- Find a function whose domain is the set of natural numbers $\mathbb{N}$ and whose range is the set of integers $\mathbb{Z}$.
Problem 34.
- Find functions $f:B\to C$, $g:A\to B$ and $h:A\to B$ such that $f\circ g=f\circ h$, yet $g\ne h$.
- Suppose $f:A\to B$ is a surjective function. Prove or disprove: If $g:B\to C$ and $h:B\to C$ satisfy $g\circ f=h\circ f$, then $g=h$.
