ATAR Notes: Forum

Uni Stuff => Science => Faculties => Mathematics => Topic started by: Yendall on March 08, 2013, 05:28:22 pm

Title: Concrete Sets - Subsets (Belonging to..)
Post by: Yendall on March 08, 2013, 05:28:22 pm: Just need some clarification for my MATH-1074 class.
We are following the rule: $\varnothing \subseteq \mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q}\subseteq \mathbb{R}$

Where:
$\varnothing$ = Null Set
$\mathbb{N}$ = Natural Numbers
$\mathbb{Z}$ = Integers
$\mathbb{Q}$ = Rational Numbers
$\mathbb{R}$ = Rational and Irrational Numbers
As far as I can understand, Null belongs to N, N belongs to Z,Z belongs to Q, Q belongs to R
So when I get a question like: To which (if any) of the sets N, Z, Q and R does 0 belong?

I automatically say that 0 is a part of the Integer set, as it is not a positive counting number and cannot belong to the natural set N. However this isn't the correct answer.
Why isn't this the correct answer?
EDIT
I realised that the multiple choice quiz online actually requires you to state that it belongs to more than one set, rather than the direct set after it.

$\therefore 0 \subseteq \mathbb{Z} \subseteq \mathbb{Q}\subseteq \mathbb{R}$

Sorry for wasting a thread.
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: kamil9876 on March 08, 2013, 07:21:53 pm: I would like to point out your confusion between $\in$ and $\subset$

Quote from: Yendall on March 08, 2013, 05:28:22 pm

$\therefore 0 \subseteq \mathbb{Z} \subseteq \mathbb{Q}\subseteq \mathbb{R}$ [/center]

It should be

$0 \in \mathbb{Z} \subseteq \mathbb{Q}\subseteq \mathbb{R}$

or the following would also be true

$\{0 \} \subset \mathbb{Z} \subseteq \mathbb{Q}\subseteq \mathbb{R}$

For instance, let $S=\{ \{1,2\}, 3\}$

Then is...

$3 \subset S$ ?

$3 \in S$ ?

$1 \in S$ ?

$\{1,2\} \subset S$ ?

$\{1,2\} \in S$ ?
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: Yendall on March 08, 2013, 10:01:40 pm: Oh yes sorry, I meant to write that. 0 is an element of each set because it is an element of set Z.

I understand elements completely, I was just confused as to whether you needed to mention ALL sets when referring to belonging.
And I believe these three are only true:

$3 \in S$

$1 \in S$

$\{1,2\} \subset S$
I would also write it as:

$X = \{1,2\}$

$X \subseteq S$
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: kamil9876 on March 09, 2013, 12:30:03 am: The elements of $S$ are $\{1,2\}$ and $3$ . So $1 \in S$ is FALSE since it is not equal to $\{1,2\}$ or $3$ . However $\{1,2\} \in S$ is TRUE, this is an element of $S$ . However $\{1,2\} \subset S$ is FALSE, there are only four subsets of $S$ : the empty set, $\{ \{1,2\} \}$ , $\{3 \}$ and $S$ itself. So you can see that $\{1,2\}$ is not one of these four sets.
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: Yendall on March 09, 2013, 07:31:23 am: Oh sorry I completely misread that $\{1,2\}$ is a seperate element to $3$ .
So;

$\{1,2\} \in S$

$3 \in S$
So the three sets in S are:

$\varnothing \in S$

$\{1,2\} \in S$

$3 \in S$
However, would this make the cardinality of the set 3 or 2? Do you count the null set as an element or because it technically isn't real, is it excluded?
Also, if I were to denote these elements with letters would this work as well:

$x = \{1,2\}$

$y = 3$
$\therefore x \subseteq S$

$\therefore y \subseteq S$
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: kamil9876 on March 09, 2013, 12:23:55 pm: Quote
However, would this make the cardinality of the set 3 or 2? Do you count the null set as an element or because it technically isn't real, is it excluded?

The cardinality is the number of elements, there are two elements $\{1,2\}$ and $3$ in $S$ hence it is of cardinality $2$ . The empty set is NOT an element of $S$ , it is a subset of $S$ i.e $\varnothing \notin S$ but $\varnothing \subset S$ . Whether it is real or not does not matter, you can make up abstract sets where the elements are not real numbers (in fact $S$ is an example of one since the element $\{1,2\}$ of $S$ is NOT a real number)

Quote from: Yendall on March 09, 2013, 07:31:23 am
Also, if I were to denote these elements with letters would this work as well:

$x = \{1,2\}$

$y = 3$
$\therefore x \subseteq S$

$\therefore y \subseteq S$

But you were doing so well before (only mistake was $\varnothing \in S$ which is FALSE). Why does renaming the elements suddenly change the answer? It shouldn't of course, you can't make $2+2=5$ by letting $2=x, 2=y, 5=z$
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: Yendall on March 09, 2013, 05:32:08 pm: Okay I think i've just completely confused myself. I'll start from the beginning again and see where I go wrong haha:
Set:

$S = \{\{1,2\}, 3\}$
Elements:

$\{1,2\} \in S$

$3 \in S$
Subsets:

$\varnothing \subseteq S$
I believe that is all there is to it! Maybe haha
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: kamil9876 on March 10, 2013, 01:38:52 am: All those statements you have made are correct. (Although there are other subsets of $S$ , in fact four altogether as I have stated in my previous post).
Title: Re: Concrete Sets - Subsets (Belonging to..)
Post by: Yendall on March 10, 2013, 09:20:09 am: Quote from: kamil9876 on March 10, 2013, 01:38:52 am
All those statements you have made are correct. (Although there are other subsets of $S$ , in fact four altogether as I have stated in my previous post).
Oh yes that's right! I think i understand it all now, thanks for your help! Sorry if I was a bit confusing haha