23666: HW 7 - Revision history

HelmutKnaust at 22:24, 30 April 2019

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HelmutKnaust at 22:18, 30 April 2019

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HelmutKnaust: Created page with "'''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 $\m..."

2019-04-30T22:16:03Z

Created page with "'''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 $\m..."

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'''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 in this order have a least upper bound?

'''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.'''
#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 $g=h$.

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-'''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 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.
@@ Line 6: / Line 6: @@
 '''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 $f=\chi_A$.
+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 $\chi_A+\chi_B=\chi_{A\cup B}$.

← Older revision		Revision as of 22:18, 30 April 2019
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−	'''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?	+	'''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.		'''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.

← Older revision		Revision as of 22:17, 30 April 2019
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−	'''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 in this order have a least upper bound?	+	'''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?

	'''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.		'''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.