Help with Set-Builder notation

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In summary, the set consisting of the values of a continuous function, G(x), over a range of integers can be expressed as G = {g(x) : xε(-range, range)} or G = {g(x): g:[-range,range] => R]}. For the vector formed by averaging m other vectors, all with n dimensions(indices), the correct expression is \mathbf{\bar{S}} = \left\{ \dfrac1m \sum\limits_{i=1}^m \mathbf{S_i}_j : \forall \left(j \in \mathbb{Z}\right)\left(j \in [1,n]\right) \right\}. Set-builder notation is
  • #1
wil3
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Hello. Say that I want to express a set consisting of the values of a continuous function, G(x), over a range of integers. Is this a valid/acceptable format?
[tex] \mathbf{G} = \{G(x) : \forall (x \in \mathbb{Z})(x \in [-range, range])\} [/tex]

or say that I want to express the vector formed by averaging m other vectors, all with n dimensions(indices). Would this express the operation correctly?

[tex]
\mathbf{\bar{S}} = \left\{ \dfrac1m \sum\limits_{i=1}^m \mathbf{S_i}_j : \forall \left(j \in \mathbb{Z}\right)\left(j \in [1,n]\right) \right\}
[/tex]

Thanks for any help!
 
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  • #2
wil3 said:
Hello. Say that I want to express a set consisting of the values of a continuous function, G(x), over a range of integers. Is this a valid/acceptable format?
[tex] \mathbf{G} = \{G(x) : \forall (x \in \mathbb{Z})(x \in [-range, range])\} [/tex]

or say that I want to express the vector formed by averaging m other vectors, all with n dimensions(indices). Would this express the operation correctly?

[tex]
\mathbf{\bar{S}} = \left\{ \dfrac1m \sum\limits_{i=1}^m \mathbf{S_i}_j : \forall \left(j \in \mathbb{Z}\right)\left(j \in [1,n]\right) \right\}
[/tex]

Thanks for any help!

For the 1st one the set builder notation is:

G = {g(x) : xε(-range, range)} or G = {g(x): g:[-range,range] => R]}

Now since the rang of g is a set of integers ,then by definition g is continuous

For the 2nd problem : Do you mean that you want to express the average sum of :

[tex]a_{1}+a_{2}+...a_{m}[/tex] vectors ,where ,

[tex] a_{1}[/tex]= ([tex] x_{11},x_{12},...x_{1n}[/tex])??
 
  • #3
So how do I make it clear that the set being defined in the first problem is only the values of G when it is given an integer argument? Ie, I want to take a continuous function and use it to generate a set by giving it only certain inputs. Think of the set I'm trying to build as some discrete pixelated approximation of G.

I want to take the average of a group of vectors by averaging their components at a series of posititions. For example, this operation of the vectors (1,3,5) and (2,4,6) would yield (1.5, 3.5,5.5)
 
  • #4
wil3 said:
So how do I make it clear that the set being defined in the first problem is only the values of G when it is given an integer argument? Ie, I want to take a continuous function and use it to generate a set by giving it only certain inputs. Think of the set I'm trying to build as some discrete pixelated approximation of G.

I want to take the average of a group of vectors by averaging their components at a series of posititions. For example, this operation of the vectors (1,3,5) and (2,4,6) would yield (1.5, 3.5,5.5)

Let us take things one by one >

G = {[tex]g(x) : x\in Z\wedge -a\leq x\leq a, g(x)=x^2[/tex]} ,where a is an integer

I just used an example of a continuous function
 
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  • #5
There are three basic ways to use set-builder notation to define a set:
  • [tex]\{ x \in A \mid P(x) \}[/tex]
  • [tex]\{ f(x) \mid x \in A \}[/tex]
  • [tex]\{ f(x) \mid x \in A \wedge P(x) \}[/tex]
where A is a set, f is a function, and P is a predicate. (all three are allowed to involve variables other than x)

Key syntactic features of these are:
  • x is introduced as a dummy variable. (Similar to how x appears in expressions like [itex]\int_a^b f(x) \, dx[/itex])
  • All forms include something that "binds" x to range over a particular set.
  • Forms 2 and 3 require an expression in x on the left side, and forms 1,3 involve a predicate in x on the right hand side

A lot of variations are allowed, so long as it's clear what is meant. (e.g. I might often write a comma instead of [itex]\wedge[/itex] in the third form, and I might omit the [itex]x \in A[/itex] part if it can be inferred from the rest of the expression)


The notation you were using is flawed because your right sides aren't expressions in the dummy variable. It's a similar grammatical error as something like:
[tex]f(x) = \int_0^1 x^2 \, dx[/tex]​
which should be avoided because the x appearing in the integral cannot be the x appearing in the left hand side.

Of course, the above equation is technically a valid one that asserts [itex]f(x) = 1/3[/itex].



I think your thinking was that you wanted to express that you were defining the set of all things described by the notation -- however that is already what the set-builder notation means. The third form, for example, reads as
The set of all f of x where x comes from A and satisfies P​

But what you wrote unfortunately introduces a new dummy variable. Your expression
[tex]\mathbf{G} = \{G(x) : \forall (x \in \mathbb{Z})(x \in [-range, range])\}[/tex]​
means exactly the same thing as
[tex]\mathbf{G} = \{G(x) : \forall (y \in \mathbb{Z})(y \in [-range, range])\}[/tex]​
Technically this is well-defined (keeping in mind the acceptable variations on notation), but it defines G to be the empty empty set, because your expression on the right hand side is identically false.
 
  • #6
Ah, I understand now. Had I not used the "for all" symbol, would my expressions be acceptable? Or do paired products of parenthetical arguments not imply logical conjunction?
 
  • #7
wil3 said:
Ah, I understand now. Had I not used the "for all" symbol, would my expressions be acceptable? Or do paired products of parenthetical arguments not imply logical conjunction?
It's fine, I think, but it really does look odd -- I expect most people would understand what you meant, even if they find it odd or think you made a typographical error.


I imagine it would be much more acceptable in certain contexts -- e.g. in a book that has already adopted the convention to write "logical and" as an implied product.
 
  • #8
Fair enough-- the notation I was using was from Wikipedia, so I'd expect a little bit of irregularity. I'll switch to the standard conjunction from now on.

I'm assuming that the vertical bar and colon are interchangeable, although please let me know if they are not for any reason. The bar reminds me a little bit too much of the "or" operator in programming, hence my avoidance of it.

Thanks very much for your help. This was all great advice.
 
  • #9
I view them as interchangeable, but I really have no idea how widespread that view is.
 
  • #10
wil3 said:
I'm assuming that the vertical bar and colon are interchangeable, although please let me know if they are not for any reason. The bar reminds me a little bit too much of the "or" operator in programming, hence my avoidance of it.

They are interchangeable; they're just different notations for the same thing.
 

Question 1: What is Set-Builder notation?

Set-Builder notation is a method used to describe a set of numbers or elements by using a rule or condition. It is written as {x | condition}, where x is the variable and the condition is the rule that each element in the set must follow.

Question 2: How do I read Set-Builder notation?

Set-Builder notation is read as "the set of all x such that x satisfies the given condition". For example, {x | x is an even number} would be read as "the set of all x such that x is an even number".

Question 3: What are the benefits of using Set-Builder notation?

Set-Builder notation allows for a concise and precise way to describe a set of numbers or elements. It also allows for the use of mathematical symbols and operations to define the set, making it easier to understand and work with.

Question 4: Can Set-Builder notation be used for infinite sets?

Yes, Set-Builder notation can be used for infinite sets. For example, {x | x is a positive integer} would represent the set of all positive integers, which is an infinite set.

Question 5: How do I write a set using Set-Builder notation?

To write a set using Set-Builder notation, start with the curly braces { } and then specify the variable followed by a vertical bar | and the condition that the elements in the set must satisfy. For example, {x | x is a prime number} would represent the set of all prime numbers.

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