Set builder notation

**novice** · May 8th 2010, 12:23 PM

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

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

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

Question:

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

**gmatt** · May 8th 2010, 01:53 PM

Originally Posted by novice

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

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

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

Question:

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

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

**Plato** · May 8th 2010, 02:04 PM

Originally Posted by gmatt

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

Did you notice that $S$ is a subset of negative integers?
That means what about $T$ ?

**novice** · May 8th 2010, 03:08 PM

Originally Posted by Plato

Did you notice that $S$ is a subset of negative integers?
That means what about $T$ ?

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

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

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

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

so $T=\{x\in \mathbb{Z}: x<0\}$ 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.

**gmatt** · May 8th 2010, 03:20 PM

Originally Posted by Plato

Did you notice that $S$ is a subset of negative integers?
That means what about $T$ ?

I'm not exactly sure what you mean, if $S \subset \mathbb{N}$ 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 $S$ , not adding another condition onto it ( since trivially if $S \subset \mathbb{N}$ and $S \subset \mathbb{Z}^-$ then $S = \emptyset$ )

**gmatt** · May 8th 2010, 03:22 PM

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

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

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

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

so $T=\{x\in \mathbb{Z}: x<0\}$ 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 $S \subset \mathbb{N}$ then $T$ are all the elements of $S$ negated...

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

then

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

**Plato** · May 8th 2010, 03:27 PM

Originally Posted by gmatt

I'm not exactly sure what you mean, if $S \subset \mathbb{N}$ then it can't be a subset of the negative integers.

Originally Posted by novice

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

Did you bother to read the OP?
It clearly states that $S$ is a set of negative integers.

**novice** · May 8th 2010, 03:29 PM

Originally Posted by gmatt

I'm not exactly sure what you mean, if $S \subset \mathbb{N}$ 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 $S$ , not adding another condition onto it ( since trivially if $S \subset \mathbb{N}$ and $S \subset \mathbb{Z}^-$ then $S = \emptyset$ )

Sorry, I didn't mean to confuse anyone. I had one thing and I asked myself a question by switching $S\subset \mathbb{Z}^-$ to $S\subset \mathbb{Z}^+$ . 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.

**novice** · May 8th 2010, 03:37 PM

Originally Posted by Plato

Did you bother to read the OP?
It clearly states that $S$ 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.

**gmatt** · May 8th 2010, 04:08 PM

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.

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