24178: HW7

Problem 31. The relation "$\preceq$" on a Boolean Algebra ${\cal B}$ defined by $A\preceq B \Leftrightarrow A\sqcup B=B$ for $A,B\in{\cal B}$ is a partial order (cp. Problem 29). Let ${\cal B}$ be a Boolean Algebra with null-element $N$, partially ordered by $\preceq$. We say that $A\in{\cal B}$ is an atom of ${\cal B}$ if $N$ is the immediate predecessor of $A$.

1. Find all atoms of ${\cal P}(\{1,2,3,4\})$.
2. Find all atoms of ${\cal D}_{42}$, defined in Problem 21.
3. Suppose the Boolean Algebra ${\cal B}$ has finitely many elements. Show that for every $B\in{\cal B}$ with $B\neq N$ there is an atom $A$ such that $A\preceq B$.

Problem 32. 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 33.

1. Show that the product of two characteristic functions is a characteristic function.
2. 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 34.

1. Find a function whose domain is the set of real numbers $\mathbb{R}$ and whose range is the set of rational numbers $\mathbb{Q}$.
2. Find a function whose domain is the set of natural numbers $\mathbb{N}$ and whose range is the set of integers $\mathbb{Z}$.

Problem 35.

1. Find functions $f:B\to C$, $g:A\to B$ and $h:A\to B$ such that $f\circ g=f\circ h$, yet $g\ne h$.
2. 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 $g=h$.