HelmutKnaust at 22:35, 30 September 2013

2013-09-30T22:35:26Z

HelmutKnaust: Created page with "'''Problem 11.''' Let $A$ be a set, and let ${\cal K}$ be a collection of sets. Show that \[A\cap(\bigcup_{B\in{\cal K}} B)=\bigcup_{B\in{\cal K}}(A\cap B).\] '''Problem 12...."

2013-09-30T17:21:53Z

Created page with "'''Problem 11.''' Let $A$ be a set, and let ${\cal K}$ be a collection of sets. Show that \[A\cap(\bigcup_{B\in{\cal K}} B)=\bigcup_{B\in{\cal K}}(A\cap B).\] '''Problem 12...."

New page

'''Problem 11.''' Let $A$ be a set, and let ${\cal K}$ be a collection of sets. Show that
\[A\cap(\bigcup_{B\in{\cal K}} B)=\bigcup_{B\in{\cal K}}(A\cap B).\]

'''Problem 12.''' Let $A$ be a proper subset of some set $U$, and let $x\in U\setminus A$.
Let ${\cal B}$ consist of all sets of the form $C\cup\{x\}$ with $C\in{\cal P}(A)$, in other words \[{\cal B}=\{C\cup\{x\}\ |\ C\in{\cal P}(A)\}.\] Show that
# ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
# ${\cal P}(A)\cap {\cal B}=\emptyset$.

'''Problem 13.''' Use the previous problem to show that ${\cal P}(A)$ has $2^n$ elements, when $A$ has $n$ elements.

'''Problem 14.''' Prove for all natural numbers $n\geq 5$: $2^n>n^2$.

'''Problem 15.''' Let $n\in\mathbb{N}$. Conjecture a formula for the expression
\[a_n=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n\cdot (n+1)}\] and prove your conjecture by induction.

← Older revision		Revision as of 22:35, 30 September 2013
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	'''Problem 15.''' Let $n\in\mathbb{N}$. Conjecture a formula for the expression		'''Problem 15.''' Let $n\in\mathbb{N}$. Conjecture a formula for the expression
	\[a_n=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n\cdot (n+1)}\] and prove your conjecture by induction.		\[a_n=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n\cdot (n+1)}\] and prove your conjecture by induction.
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CRN 12107: HW 3 - Revision history

HelmutKnaust at 22:35, 30 September 2013

HelmutKnaust: Created page with "'''Problem 11.''' Let $A$ be a set, and let ${\cal K}$ be a collection of sets. Show that \[A\cap(\bigcup_{B\in{\cal K}} B)=\bigcup_{B\in{\cal K}}(A\cap B).\] '''Problem 12...."