HelmutKnaust: Created page with "'''Problem 31.''' On the set of natural numbers $\mathbb{N}$ consider the partial order \[n\ |\ m \Leftrightarrow\ n\mbox{ is a divisor of }m.\] #Draw a ''Hasse diagram'' fo..."

2013-11-26T06:14:49Z

Created page with "'''Problem 31.''' On the set of natural numbers $\mathbb{N}$ consider the partial order \[n\ |\ m \Leftrightarrow\ n\mbox{ is a divisor of }m.\] #Draw a ''Hasse diagram'' fo..."

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'''Problem 31.''' On the set of natural numbers $\mathbb{N}$ consider the partial order \[n\ |\ m \Leftrightarrow\ n\mbox{ is a divisor of }m.\]
#Draw a ''Hasse diagram'' for the set $A=\{1,2,3,4,5,\ldots, 12,13,14,15\}$ endowed with this partial order.
#Find the largest element (maximum) of $A$, or show that it does not exist.
#Find the maximal elements of $A$, or show that none exist.
#Find three upper bounds for $A$ in $\mathbb{N}$.

'''Problem 32.''' Consider $\mathbb{R}^2$ with the lexicographical order $\leq_\ell$: \[(a,b)\leq_\ell (c,d)\quad\Leftrightarrow\quad (a<c) \vee ((a=c) \wedge (b\leq d)).\]
#Show that $\leq_\ell$ is a linear order on $\mathbb{R}^2$.
#Find a subset of $(\mathbb{R}^2,\leq_\ell)$ that is bounded from above, but fails to have a least upper bound.

'''Problem 33.''' 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 30''').
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 20'''.
#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 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 $g=h$.

CRN 12107: HW 7 - Revision history

HelmutKnaust: Created page with "'''Problem 31.''' On the set of natural numbers $\mathbb{N}$ consider the partial order \[n\ |\ m \Leftrightarrow\ n\mbox{ is a divisor of }m.\] #Draw a ''Hasse diagram'' fo..."