Set builder notation for points on a curve

[Chris1901]

I'm doing a bit of my own learning so I have no one else to ask.
I am doing set builder notation and came across some examples that break each bit up and slightly explain each one.
There is one part of one of the examples that I'm not sure what it means.

The example is...
S = {(x,x³) ∈ ℝ²
This translates as the set of S contains all the points of the form (x,x³).
In other words it is the curve y = x³

It then breaks it down to state what each part means.
It says that (x,x³) is
all the points of the form (x,x³)
This is the part that I don't know what it means, can any body explain further, hopefully I have given enough information on what I want to find out.

Cheers,
Chris.

 

[JeffM]

I'm doing a bit of my own learning so I have no one else to ask.
I am doing set builder notation and came across some examples that break each bit up and slightly explain each one.
There is one part of one of the examples that I'm not sure what it means.

The example is...
S = {(x,x³) ∈ ℝ²
This translates as the set of S contains all the points of the form (x,x³).
In other words it is the curve y = x³

It then breaks it down to state what each part means.
It says that (x,x³) is
all the points of the form (x,x³)
This is the part that I don't know what it means, can any body explain further, hopefully I have given enough information on what I want to find out.

Cheers,
Chris.

One way to specify an ordered n-tuple of numbers is \(\displaystyle (a_1,\ ... \ a_n).\)

So \(\displaystyle (x,\ x^3)\) means an ordered pairs of numbers that have
that relationship, such as (-2, - 8) or (3, 27).

If you were to graph that set using the conventional labels on a Cartesian plane, the corresponding equation would be

\(\displaystyle y = x^3.\)

I admit that it is confusing if you are used to normal algebraic notation. It might be clearer (though less concise) to write:

\(\displaystyle \mathbb S = \{(x,\ f(x) \} \in \mathbb R^2 \text { such that } f(x) = x^3.\)

 

[pka]

I'm doing a bit of my own learning so I have no one else to ask.
I am doing set builder notation and came across some examples that break each bit up and slightly explain each one.
There is one part of one of the examples that I'm not sure what it means.
The example is... S = {(x,x³) ∈ ℝ²
This translates as the set of S contains all the points of the form (x,x³).
In other words it is the curve y = x³

It then breaks it down to state what each part means.
It says that (x,x³) is all the points of the form (x,x³)
This is the part that I don't know what it means, can any body explain further, hopefully I have given enough information on what I want to find out.

I may be reading too much into your post. If so please ignore.
Hope that you know that \(\displaystyle \{ a,b\} = \{ b,a\} \) i.e. sets are not ordered. But we are in need of ordered pairs. But if sets are the foundational building blocks of mathematics how do we define ordered pairs as un-ordered sets. Believe it or not it took over twenty to find the rather simple solution. Today we define the ordered pair \(\displaystyle (a,b)=\{\{a\},\{a,b\}\}\). That is a set of two sets, one is a single set of the first term and the other set of two elements, the first term & the second term. Thus the set \(\displaystyle \{\{6,3\},\{3\}\}\) is the ordered pair \(\displaystyle (3,6)\).

So in your example the function \(\displaystyle f(x)=x^3\) can be written as \(\displaystyle f=\{(x,x^3) : x\in\mathbb{R}\}=\{\{\{x\},\{x,x^3\}\} : x\in\mathbb{R}\}\)

 

[Chris1901]

Thanks to both of you for your replies.
Both very helpful and indeed insightful
Cheers.