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Fill in the black using $\subseteq$ OR $\not \subset $

Are my answers correct?

{$\frac {x} {x+1} : x\in \mathbb {N}$} __ $\mathbb {Q}$

ANSWER: $\not \subset $

$\mathbb {Z} \cup [-1,1]$ __ $[-2,2]

ANSWER: $\subseteq$

$\mathbb {N} $ x $\mathbb {Z}$ __ $\mathbb {Z} $ x $\mathbb {N}$

ANSWER: $\not \subset $

$\mathbb {Z} $ x $\mathbb {N}$ __ $\mathbb {N} $ x $\mathbb {Z}$

ANSWER: $\not \subset $

user114726
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1 Answers1

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The last two are correct: $\langle 1,-1\rangle$ is in $\Bbb N\times\Bbb Z$ but not in $\Bbb Z\times\Bbb N$, and $\langle -1,1\rangle$ is in $\Bbb Z\times\Bbb N$ but not in $\Bbb N\times\Bbb Z$. The first two, however, are wrong.

  • If $x\in\Bbb N$, then $x+1\in\Bbb N$ and $x+1\ne 0$, so $\frac{x}{x+1}\in\Bbb Q$. Thus, $\left\{\frac{x}{x+1}:x\in\Bbb N\right\}\subseteq\Bbb Q$.

  • $3\in\Bbb Z\cup[-1,1]$, but $3\notin[-2,2]$, so $\Bbb Z\cup[-1,1]\nsubseteq[-2,2]$.

Brian M. Scott
  • 616,228
  • @user113609: You’re welcome. – Brian M. Scott Dec 09 '13 at 23:48
  • WOW just saw your profile and you are person with best score in mse. i wish one day i could be as smart as you. btw congrats on being the number 1, must it feel like heaven being the best in mse?? –  Dec 09 '13 at 23:53
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    @user113609: Thanks! I have an unfair advantage, though: being retired, I’ve more time to spend here than most other users. – Brian M. Scott Dec 10 '13 at 00:01