24178: HW 2 - Revision history

HelmutKnaust at 19:40, 2 February 2017

2017-02-02T19:40:54Z

HelmutKnaust at 19:40, 2 February 2017

2017-02-02T19:40:17Z

HelmutKnaust: Created page with "'''Problem 6.''' Is the statement $\quad\exists !\,x\in\mathbb{R} : (x-2=\sqrt{x+7}) \quad$ true or false? Prove your conjecture. '''Problem 7.''' Let $A,B$ and $C$ be arbitr..."

2017-02-02T19:35:57Z

Created page with "'''Problem 6.''' Is the statement $\quad\exists !\,x\in\mathbb{R} : (x-2=\sqrt{x+7}) \quad$ true or false? Prove your conjecture. '''Problem 7.''' Let $A,B$ and $C$ be arbitr..."

New page

'''Problem 6.''' Is the statement $\quad\exists !\,x\in\mathbb{R} : (x-2=\sqrt{x+7}) \quad$ true or false?
Prove your conjecture.

'''Problem 7.''' 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 8.''' 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 9.''' 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 10.''' Critique the following proof. Is the proof correct or flawed? Explain!

Recall that a positive integer $p$ is ''prime'' if it is divisible by exactly two positive integers, namely $1$ and $p$. The five smallest primes are 2,3,5,7,11.

'''Theorem.''' There are infinitely many primes.

'''Proof:''' Suppose there are only finitely many primes, say the list of all primes is $\{p_1,p_2,p_3,\ldots, p_n\}$ for some positive integer $n$. Set
\[p=1+p_1\cdot p_2 \cdot p_3 \cdots p_n.\]
Then $p$ leaves a remainder of 1 when divided by any of the $p_n$'s and thus $p$ must be a prime not on the list of all primes.

[[Image:Alice3.gif]]

@@ Line 8: / Line 8: @@
 '''Problem 9.'''  Let $x$ be a natural number. Prove or disprove: If $x^2$ is divisible by 27, then $x$ is divisible by 9.
-'''Problem 10.''' Let $n$ be a natural number. Show that $\sqrt{n} is a natural number if and only if $n=k^2$ for some natural number $k$.
+'''Problem 10.''' Let $n$ be a natural number. Show that $\sqrt{n}$ is a natural number if and only if $n=k^2$ for some natural number $k$.
 [[Image:Alice3.gif]]

@@ Line 2: / Line 2: @@
 Prove your conjecture.
-'''Problem 7.''' Let $A,B$ and $C$ be arbitrary sets. Recall that $A\setminus B=\{x \ |\ x\in A\ \wedge\ x\not\in B\}$. We define
+'''Problem 7.''' Let $x$, $y$, and $z$ be natural numbers. Prove or disprove: If $x+y$ is even and $y+z$ is even, then $x+z$ is even.
-$A\bigtriangleup B:=(A\setminus B)\cup(B \setminus A)$.
 Prove or disprove:
-#$A \bigtriangleup B= B \bigtriangleup A$.
-#$(A \bigtriangleup B)\bigtriangleup C=A \bigtriangleup (B \bigtriangleup C)$.
-'''Problem 8.''' Let  $A$ and $B$ be arbitrary sets. Prove or disprove the following power set relations:
+'''Problem 8.''' Let $x$, $y$, and $z$ be natural numbers. Prove or disprove: If $x+y$ is odd and $y+z$ is odd, then $x+z$ is odd.
-#${\cal P}(A\cap B)\subseteq {\cal P}(A)\cap {\cal P}(B)$.
-#${\cal P}(A)\cap {\cal P}(B)\subseteq {\cal P}(A\cap B)$.
-'''Problem 9.'''  Given two real numbers $a<b$, the open interval $(a,b)$ is defined to be the set $\displaystyle{\{x\in\mathbb{R}\ |\ (a<x) \wedge (x<b)\}}$.
+'''Problem 9.'''  Let $x$ be a natural number. Prove or disprove: If $x^2$ is divisible by 27, then $x$ is divisible by 9.
-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 10.''' Let $n$ be a natural number. Show that $\sqrt{n} is a natural number if and only if $n=k^2$ for some natural number $k$.
-−
-−
-'''Problem 10.''' Critique the following proof. Is the proof correct or flawed? Explain!
-−
-Recall that a positive integer $p$ is ''prime'' if it is divisible by exactly two positive integers, namely $1$ and $p$. The five smallest primes are 2,3,5,7,11.
-−
-'''Theorem.''' There are infinitely many primes.
-−
-'''Proof:''' Suppose there are only finitely many primes, say the list of all primes is $\{p_1,p_2,p_3,\ldots, p_n\}$ for some positive integer $n$. Set
-\[p=1+p_1\cdot p_2 \cdot p_3 \cdots p_n.\]
-Then $p$ leaves a remainder of 1 when divided by any of the $p_n$'s and thus $p$ must be a prime not on the list of all primes.
 [[Image:Alice3.gif]]