# Thread: Set builder notation

1. ## Set builder notation

In the beginning part of my book. There was a solution which says:

Given a set of negative integers, $\displaystyle S$
, where $\displaystyle S \subset \mathbb{Z}^-$, we can build a set $\displaystyle T$ of positive numbers by a set builder notation such as

$\displaystyle T=\{x \in Z: -x \in S\}$.

Question:

Since
$\displaystyle x$ is only a dummy variable, I am wondering whether it's possible at all to built the set $\displaystyle T$ with $\displaystyle S \subset \mathbb{N}$?

2. Originally Posted by novice
In the beginning part of my book. There was a solution which says:

Given a set of negative integers, $\displaystyle S$
, where $\displaystyle S \subset \mathbb{Z}^-$, we can build a set $\displaystyle T$ of positive numbers by a set builder notation such as

$\displaystyle T=\{x \in Z: -x \in S\}$.

Question:

Since
$\displaystyle x$ is only a dummy variable, I am wondering whether it's possible at all to built the set $\displaystyle T$ with $\displaystyle S \subset \mathbb{N}$?

You can, but then T would be a subset of the negative integers.

3. Originally Posted by gmatt
You can, but then T would be a subset of the negative integers.
Did you notice that $\displaystyle S$ is a subset of negative integers?
That means what about $\displaystyle T$?

4. Originally Posted by Plato
Did you notice that $\displaystyle S$ is a subset of negative integers?
That means what about $\displaystyle T$?
I found that set notations can be very tricky. I have many dumb questions. Apparently, they also teach me the most. It made me think.

I am thinking: Since elements of $\displaystyle S$ are negative, where $\displaystyle S \subset \mathbb{Z}^-$, and that $\displaystyle -x\in S$, then $\displaystyle x$ ought to be positive, i.e. $\displaystyle x>0$, so $\displaystyle T=\{x\in Z: x>0\}$.

I contemplated about the question you asked--and of course, the poster immediately before you too helped me think. So, I wrote on a piece of paper

If $\displaystyle S\not= \emptyset$ and $\displaystyle S \subset \mathbb{N}$, then the elements of $\displaystyle S$ cannot be negative, despite $\displaystyle T$ being defined as

$\displaystyle T=\{x\in \mathbb{Z}: -x\in S\}$. Since

$\displaystyle -x >0$, it has to be nothing else but $\displaystyle x<0$,

so $\displaystyle T=\{x\in \mathbb{Z}: x<0\}$ as gmatt said.

I am a little too slow, but I will have another 70 years (I meant 70 more years) to learn.

Thanks to both of you.

5. Originally Posted by Plato
Did you notice that $\displaystyle S$ is a subset of negative integers?
That means what about $\displaystyle T$?
I'm not exactly sure what you mean, if $\displaystyle S \subset \mathbb{N}$ then it can't be a subset of the negative integers. I assumed what the OP meant was that he was switching the meaning of $\displaystyle S$, not adding another condition onto it ( since trivially if $\displaystyle S \subset \mathbb{N}$ and $\displaystyle S \subset \mathbb{Z}^-$ then $\displaystyle S = \emptyset$)

6. Originally Posted by novice
I found that set notations can be very tricky. I have many dumb questions. Apparently, they also teach me the most. It made me think.

I am thinking: Since elements of $\displaystyle S$ are negative, where $\displaystyle S \subset \mathbb{Z}^-$, and that $\displaystyle -x\in S$, then $\displaystyle x$ ought to be positive, i.e. $\displaystyle x>0$, so $\displaystyle T=\{x\in Z: x>0\}$.

I contemplated about the question you asked--and of course, the poster immediately before you too helped me think. So, I wrote on a piece of paper

If $\displaystyle S\not= \emptyset$ and $\displaystyle S \subset \mathbb{N}$, then the elements of $\displaystyle S$ cannot be negative, despite $\displaystyle T$ being defined as

$\displaystyle T=\{x\in \mathbb{Z}: -x\in S\}$. Since

$\displaystyle -x >0$, it has to be nothing else but $\displaystyle x<0$,

so $\displaystyle T=\{x\in \mathbb{Z}: x<0\}$ as gmatt said.

I am a little too slow, but I will have another 70 years to learn.

Thanks to both of you.
Well to be clear, if $\displaystyle S \subset \mathbb{N}$ then $\displaystyle T$ are all the elements of $\displaystyle S$ negated...

i.e.
if
$\displaystyle S = \{ s_1, s_2, ... \} \subset \mathbb{N}$

then

$\displaystyle T = \{ -s_1, -s_2, ... \} \subset \mathbb{Z}^-$

7. Originally Posted by gmatt
I'm not exactly sure what you mean, if $\displaystyle S \subset \mathbb{N}$ then it can't be a subset of the negative integers.
Originally Posted by novice
Given a set of negative integers, $\displaystyle S$, where $\displaystyle S \subset \mathbb{Z}^-$, we can build a set $\displaystyle T$ of positive numbers by a set builder notation such as
$\displaystyle T=\{x \in Z: -x \in S\}$
Did you bother to read the OP?
It clearly states that $\displaystyle S$ is a set of negative integers.

8. Originally Posted by gmatt
I'm not exactly sure what you mean, if $\displaystyle S \subset \mathbb{N}$ then it can't be a subset of the negative integers. I assumed what the OP meant was that he was switching the meaning of $\displaystyle S$, not adding another condition onto it ( since trivially if $\displaystyle S \subset \mathbb{N}$ and $\displaystyle S \subset \mathbb{Z}^-$ then $\displaystyle S = \emptyset$)
Sorry, I didn't mean to confuse anyone. I had one thing and I asked myself a question by switching $\displaystyle S\subset \mathbb{Z}^-$ to $\displaystyle S\subset \mathbb{Z}^+$. Experimenting with things by taking off the head and tail, and putting tail where the head was and head where the tail was and see what shape I would get. I am groping like a blind man.

9. Originally Posted by Plato
Did you bother to read the OP?
It clearly states that $\displaystyle S$ is a set of negative integers.
Sorry, sir, I wasn't ignoring your question. I realized that I have confused you. I should have made it more clear that I was switching things around for experiment.

While I was switching things around, I confused myself and then asked question, but I have learned quite a bit though.

If there is any consolation, you do help me learn.

10. Originally Posted by novice
Sorry, sir, I wasn't ignoring your question. I realized that I have confused you. I should have made it more clear that I was switching things around for experiment.

While I was switching things around, I confused myself and then asked question, but I have learned quite a bit though.

If there is any consolation, you do help me learn.
Yes, that is the way I interpreted things, I think Plato might not have realized that the first part of your post was a statement and the later part of it was the question.