21495: HW 3 - Revision history

HelmutKnaust at 21:42, 2 March 2017

2017-03-02T21:42:35Z

HelmutKnaust at 00:39, 2 March 2017

2017-03-02T00:39:34Z

HelmutKnaust at 20:08, 20 February 2017

2017-02-20T20:08:23Z

HelmutKnaust: Created page with "'''Problem 11.''' Let $A,B$ and $C$ be arbitrary sets. Recall that $A\setminus B=\{x \ |\ x\in A\ \wedge\ x\not\in B\}$. We define $A\bigtriangleup B:=(A\setminus B)\cup(B \s..."

2017-02-20T20:06:33Z

Created page with "'''Problem 11.''' Let $A,B$ and $C$ be arbitrary sets. Recall that $A\setminus B=\{x \ |\ x\in A\ \wedge\ x\not\in B\}$. We define $A\bigtriangleup B:=(A\setminus B)\cup(B \s..."

New page

'''Problem 11.''' Let $A,B$ and $C$ be arbitrary sets. Recall that $A\setminus B=\{x \ |\ x\in A\ \wedge\ x\not\in B\}$. We define
$A\bigtriangleup B:=(A\setminus B)\cup(B \setminus A)$.
Prove or disprove:
#$A \bigtriangleup B= B \bigtriangleup A$.
#$(A \bigtriangleup B)\bigtriangleup C=A \bigtriangleup (B \bigtriangleup C)$.

'''Problem 12.''' Let $A$ and $B$ be arbitrary sets. Prove or disprove the following power set relations:
#${\cal P}(A\cap B)\subseteq {\cal P}(A)\cap {\cal P}(B)$.
#${\cal P}(A)\cap {\cal P}(B)\subseteq {\cal P}(A\cap B)$.

'''Problem 13.''' Given two real numbers $a<b$, the open interval $(a,b)$ is defined to be the set $\displaystyle{\{x\in\mathbb{R}\ |\ (a<x) \wedge (x<b)\}}$.

For $n\in\mathbb{N}$, let $A_n$ be the open interval $\displaystyle{(\frac{1}{2}-\frac{1}{2n}, \frac{1}{2}+\frac{1}{3n})}$. Find $\displaystyle{\bigcup_{n\in\mathbb{N}} A_n}$ and $\displaystyle{\bigcap_{n\in\mathbb{N}} A_n}$. Confirm your conjectures by proofs.

@@ Line 4: / Line 4: @@
 #$A \bigtriangleup B= B \bigtriangleup A$.
 #$(A \bigtriangleup B)\bigtriangleup C=A \bigtriangleup (B \bigtriangleup C)$.
 '''Problem 12.''' Let  $A$ and $B$ be arbitrary sets. Prove or disprove the following power set relations:
@@ Line 20: / Line 22: @@
 # ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
 # ${\cal P}(A)\cap {\cal B}=\emptyset$.

← Older revision		Revision as of 00:39, 2 March 2017
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	# ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.		# ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
	# ${\cal P}(A)\cap {\cal B}=\emptyset$.		# ${\cal P}(A)\cap {\cal B}=\emptyset$.
		+
		+	[[Image:assocSSD.png\|frame\|center\|Problem 11.2]]

← Older revision		Revision as of 20:08, 20 February 2017
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	For $n\in\mathbb{N}$, let $A_n$ be the open interval $\displaystyle{(\frac{1}{2}-\frac{1}{2n}, \frac{1}{2}+\frac{1}{3n})}$. Find $\displaystyle{\bigcup_{n\in\mathbb{N}} A_n}$ and $\displaystyle{\bigcap_{n\in\mathbb{N}} A_n}$. Confirm your conjectures by proofs.		For $n\in\mathbb{N}$, let $A_n$ be the open interval $\displaystyle{(\frac{1}{2}-\frac{1}{2n}, \frac{1}{2}+\frac{1}{3n})}$. Find $\displaystyle{\bigcup_{n\in\mathbb{N}} A_n}$ and $\displaystyle{\bigcap_{n\in\mathbb{N}} A_n}$. Confirm your conjectures by proofs.
		+
		+	'''Problem 14.''' 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 15.''' 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
		+	# ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
		+	# ${\cal P}(A)\cap {\cal B}=\emptyset$.