Building a set notation part II

In summary, the attempt at a solution is to use Z rather than N to define the set, and to use properties of x rather than using two properties of x. This method is accepted, but using two properties of x is not required.
  • #1
reenmachine
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Homework Statement



As an exercise , the book I'm reading ask me to build a set notation for the following set:

##\{... \ , \frac{1}{27} \ , \frac{1}{9} \ , \frac{1}{3} \ , 1 \ , 3 \ , 9 \ , 27 , \ ...\}##

The Attempt at a Solution



After playing with the numbers a couple of minutes , I came with this result:

##\{ x \in R : \exists y \in N \ \ 3^y = x \ \ \ 1/(3^y) = x\}##

Here I'm wondering if the right side is correct.The reason for my doubts is the fact I used two properties of x instead of one.

thanks!
 
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  • #2
Attempt at describing the set using another road:

##\{ x \in \{3^y\}\cup\{1/(3^y)\} : y \in N \}##

Is this accepted? Or is making a statement about y on the right side instead of x disqualifies this method?

edit:

##\{ x \in R : \exists y \in N \ \ x \in \{3^y\}\cup\{1/(3^y)\}\}##
 
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  • #3
reenmachine said:
Attempt at describing the set using another road:

##\{ x \in \{3^y\}\cup\{1/(3^y)\} : y \in N \}##

Is this accepted? Or is making a statement about y on the right side instead of x disqualifies this method?

1 is also supposed to be an element of your set. Is it? And you still aren't describing it as directly as you could.
 
  • #4
reenmachine said:

Homework Statement



As an exercise , the book I'm reading ask me to build a set notation for the following set:

##\{... \ , \frac{1}{27} \ , \frac{1}{9} \ , \frac{1}{3} \ , 1 \ , 3 \ , 9 \ , 27 , \ ...\}##

The Attempt at a Solution



After playing with the numbers a couple of minutes , I came with this result:

##\{ x \in R : \exists y \in N \ \ 3^y = x \ \ \ 1/(3^y) = x\}##

Here I'm wondering if the right side is correct.The reason for my doubts is the fact I used two properties of x instead of one.

thanks!
Almost! The (modern) standard definition of "N" is that it is the set of positive integers and so does not include 0 which means your set does not include [tex]3^0= 1[/tex].

Think about using "I" (the set of all integers) instead of N.
 
  • #5
oops , forget about it , brain cramp

I was unaware that ##3^0 = 1## to be honest.

I thought it gave us 0 and that ##1/(3^y) = 1## if ##y=0##.

But I guess the fact that I thought it would give us 0 was not good since 0 isn't part of the set.
 
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  • #6
reenmachine said:
oops , forget about it , brain cramp

I was unaware that ##3^0 = 1## to be honest.

I thought it gave us 0 and that ##1/(3^y) = 1## if ##y=0##.

But I guess the fact that I thought it would give us 0 was not good since 0 isn't part of the set.

Once you've got the powers straightened out, why don't you try to use Z instead of N, like in the last post. Remember 3^(-k)=1/3^k?
 
  • #7
##\{ x \in R : \exists y \in Z \ \ 3^y=x \}##
 
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  • #8
Dick said:
Once you've got the powers straightened out, why don't you try to use Z instead of N, like in the last post. Remember 3^(-k)=1/3^k?

Yeah this is what I did but I had to refresh my mind about negative exponants.This made me see the problem more complicated than it was.

thanks man!
 
  • #9
Edit: I clicked the quote button and then left the computer for a while. Didn't see that a lot had happened since then.

Edit 2: Oh, and I also overlooked the issue with 0 that HallsofIvy mentioned.

reenmachine said:

Homework Statement



As an exercise , the book I'm reading ask me to build a set notation for the following set:

##\{... \ , \frac{1}{27} \ , \frac{1}{9} \ , \frac{1}{3} \ , 1 \ , 3 \ , 9 \ , 27 , \ ...\}##

The Attempt at a Solution



After playing with the numbers a couple of minutes , I came with this result:

##\{ x \in R : \exists y \in N \ \ 3^y = x \ \ \ 1/(3^y) = x\}##

Here I'm wondering if the right side is correct.The reason for my doubts is the fact I used two properties of x instead of one.

thanks!
If you want to use two properties P(x) and Q(x), you need to make it clear if you mean "P(x) and Q(x)", "P(x) or Q(x)", or something else.

In this case, you could type \text{ or }. But a better approach is to replace ##\mathbb N## with ##\mathbb Z##. You know that ##x^{-y}=1/x^y## for all ##x,y\in\mathbb R##, right?
reenmachine said:
Attempt at describing the set using another road:

##\{ x \in \{3^y\}\cup\{1/(3^y)\} : y \in N \}##

Is this accepted? Or is making a statement about y on the right side instead of x disqualifies this method?
When you read it as "the set of all x in ##\{3^y\}\cup\{1/(3^y)\}## such that..." you should see that this only makes sense if y is some specific number defined earlier. In that case, your notation defines a subset of a set with only two elements.

reenmachine said:
edit:

##\{ x \in R : \exists y \in N \ \ x \in \{3^y\}\cup\{1/(3^y)\}\}##
This is logically correct, because ##x \in \{3^y\}\cup\{1/(3^y)\}## means $$x=3^y\text{ or }x=1/3^y,$$ but the notation is kind of ugly.
 
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  • #10
reenmachine said:
##\{ x \in R : \exists y \in Z \ \ 3^y=x \}##

Better, but still overcomplicated. What's wrong with ##\{3^y : y \in Z \}##?
 
  • #11
HallsofIvy said:
Almost! The (modern) standard definition of "N" is that it is the set of positive integers and so does not include 0 which means your set does not include [tex]3^0= 1[/tex].

Think about using "I" (the set of all integers) instead of N.
The book reenmachine has been using defines ##\mathbb N## that way, but I don't think there's a standard. I know I prefer to include 0.
 
  • #12
Dick said:
Better, but still overcomplicated. What's wrong with ##\{3^y : y \in Z \}##?

Not sure.

It's just that when I'm saying it verbally , like ''the set of all ##3^y## such that ##y \in Z##'' sounds a little bit weirder than ''the set of all ##x \in R## such that ##3^y = x## if there exist a ##y \in Z''##.

But ##\{3^y : y \in Z \}## is the shortest answer so you're right.
 
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  • #13
reenmachine said:
Not sure.

It's just that when I'm saying it verbally , like ''the set of all ##3^y## such that ##y \in Z##'' sounds a little bit weirder than ''the set of all ##x \in R## such that ##3^y = x## if there exist a ##y \in Z''##.

But ##\{3^y : y \in Z \}## is the shortest answer so you're right.

Your way isn't wrong and the shortest way isn't necessarily right, but once you get used to it, I think shorter way is easier to read.
 

FAQ: Building a set notation part II

1. What is set notation and why is it important in building a mathematical set?

Set notation is a symbolic representation used to describe and define a mathematical set. It is important because it allows us to clearly and concisely communicate the elements and properties of a set, making it easier to understand and manipulate in mathematical operations.

2. How do you use intervals in set notation?

An interval is a range of values between two endpoints. In set notation, intervals are represented using brackets or parentheses. For example, [a,b] represents the set of all real numbers between a and b, including a and b, while (a,b) represents all real numbers between a and b, but excluding a and b.

3. Can you use set-builder notation to describe infinite sets?

Yes, set-builder notation can be used to describe infinite sets. For example, the set of all even numbers can be written as {x | x is an even number}, where the vertical bar "|" means "such that". This notation allows us to describe an infinite number of elements in a set without having to list them all individually.

4. How are set operations represented in set notation?

Set operations, such as union, intersection, and complement, are represented using symbols. For example, the union of two sets A and B is written as A ∪ B, the intersection is written as A ∩ B, and the complement of a set A is written as A̅.

5. Are there any other notations used in set theory?

Yes, besides set-builder notation and interval notation, there are other notations used in set theory. These include symbolic notation, which uses symbols to represent elements and relationships in a set, and Venn diagrams, which use overlapping circles to visually represent set relationships.

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