View source for 21495: 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 $A$.
#Find all atoms of ${\cal P}(\{1,2,3,4\})$.
#Find all atoms of ${\cal D}_{42}$, defined in '''Problem 21'''.
#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.'''
#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 34.'''  
#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 35.''' 
#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 $g=h$.