set builder problem

• Jan 28th 2010, 05:29 AM
canyiah
set builder problem
i cant figure out if how to write this in set builder notation
{{0},{0,1},{0,1,2},{0,1,2,3}.....} can anyone help me with this i guessed that you could do it like this
{x | for all x there is a value y that is y < x i and y and x are positive integers}
• Jan 28th 2010, 05:43 AM
Dinkydoe
There's probably a few ways to do it, I'd say:

$\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): x\in u \wedge y< x\rightarrow y\in u \right\}$

I think this will do it
• Jan 28th 2010, 10:27 AM
novice
Quote:

Originally Posted by Dinkydoe
There's probably a few ways to do it, I'd say:

$\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): x\in u \wedge y< x\rightarrow y\in u \right\}$

I think this will do it

Is it $\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): (x\in u) \wedge (y< x)\rightarrow y\in u \right\}$?

$\displaystyle (y< x)$ part is hard to picture.

Can we say $\displaystyle \{u\in \mathcal{P}(\mathbb{N}): u =[0,n] , n \in Z^{+}\}$?
• Jan 28th 2010, 11:03 AM
Dinkydoe
That's true. It's the same. I believe setting these "()" is not strictly necessary, but it makes it better readible I guess ;)

Quote:
That's a good question. Actually I'm not sure: Normally we keep the notation $\displaystyle [a,b]$ as an interval in $\displaystyle \mathbb{Q}$, or $\displaystyle \mathbb{R}$.

Anyway: I think just writing $\displaystyle \left\{\left\{0,1,\cdots ,n\right\}: n\in \mathbb{N}\right\}$ Is suggestive enough in itself. Every mathematician will know what you mean ;p

$\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): (x\in u) \wedge (y< x)\rightarrow y\in u \right\}$ is a more formal notation.
• Jan 28th 2010, 11:26 AM
novice
Quote:

Originally Posted by Dinkydoe
That's true. It's the same. I believe setting these "()" is not strictly necessary, but it makes it better readible I guess ;)

That's a good question. Actually I'm not sure: Normally we keep the notation $\displaystyle [a,b]$ as an interval in $\displaystyle \mathbb{Q}$, or $\displaystyle \mathbb{R}$.

Anyway: I think just writing $\displaystyle \left\{\left\{0,1,\cdots ,n\right\}: n\in \mathbb{N}\right\}$ Is suggestive enough in itself. Every mathematician will know what you mean ;p

$\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): (x\in u) \wedge (y< x)\rightarrow y\in u \right\}$ is a more formal notation.

Let's give another try:

Is $\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): 0\leq x\leq n, x \in u, n\in \mathbb{N} \right\}$ considered a formal notation?

In this lesson, I have learned from you "Normally we keep the notation $\displaystyle [a,b]$ as an interval in $\displaystyle \mathbb{Q}$, or $\displaystyle \mathbb{R}$."(Happy)
• Jan 28th 2010, 11:49 AM
canyiah
thanks for the quick replys I am still waiting for my prof to tell me the solution proved was correct here was his sugguestions prior to me posting this on the board. Im guessing this is going to be on the quiz since he wont just tell me the answer.

----
What is the set { x | 0 <= x <= b } ? Isn't that the typical element of the set of sets we are trying to describe? Now just index this over all the relevant values of b, using appropriate set-builder notation.
----
include all the integers from 0 to k for each fixed k. Find a way to

>> say that using set-builder notation (you'll be using set-builder

>> notation within set-builder notation).
• Jan 28th 2010, 08:25 PM
novice
For your question, I think it's simpler to write $\displaystyle \{A_{n \in \mathbb{N}}\}=\{\{0\},\{0,1\},\{0,1,2\},...\}$, so that in set builder form

we can write $\displaystyle \{A_{n \in \mathbb{N}}\}$, where $\displaystyle A_n= \{a\in \mathbb{N}: 0\leq a \leq (n-1) \}$.
• Jan 28th 2010, 08:38 PM
Drexel28
Quote:

Originally Posted by novice
Is it $\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): (x\in u) \wedge (y< x)\rightarrow y\in u \right\}$?

$\displaystyle (y< x)$ part is hard to picture.

Can we say $\displaystyle \{u\in \mathcal{P}(\mathbb{N}): u =[0,n] , n \in Z^{+}\}$?

It is common notation to denote $\displaystyle \left\{1,\cdots,n\right\}$ by either $\displaystyle \bold{n}$ or $\displaystyle [n]$. Make sure you specify. Also, it is incorrectly
• Jan 29th 2010, 06:19 AM
novice
Quote:

Originally Posted by Drexel28
It is common notation to denote $\displaystyle \left\{1,\cdots,n\right\}$ by either $\displaystyle \bold{n}$ or $\displaystyle [n]$. Make sure you specify. Also, it is incorrectly

You are right, now that I am learning theoritical math, I need to form a new habit of specifying notations.

I am so glad you metioned the notations in quote. I have never seen them before.