23666: HW 3 - Revision history

HelmutKnaust at 19:28, 27 February 2019

2019-02-27T19:28:42Z

HelmutKnaust at 19:28, 27 February 2019

2019-02-27T19:28:14Z

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..."

2019-02-27T19:26:07Z

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..."

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'''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:
#$A \bigtriangleup B= B \bigtriangleup A$.
#$(A \bigtriangleup B)\bigtriangleup C=A \bigtriangleup (B \bigtriangleup C)$.

[[Image:assocSSD.png|frame|center|Problem 11.2]]

'''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.

'''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
# ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
# ${\cal P}(A)\cap {\cal B}=\emptyset$.

@@ Line 6: / Line 6: @@
 <!--[[Image:assocSSD.png|frame|center|Problem 11.2]]-->
 '''Problem 12.''' Let  $A$ and $B$ be arbitrary sets. Prove or disprove the following power set relations:
 #${\cal P}(A\cap B)\subseteq {\cal P}(A)\cap {\cal P}(B)$.