Describing Sets: A Comprehensive Guide

[mathmari]

Hey!

• I want to describe in words the following sets:
  
  1. $A:=\{(x,y)\in \mathbb{R}^2\mid x>0, y\leq 1\}$
  
  $A$ is the set of all pointgs where the first coordinate is positiv and the second one is less or equal to $1$.
  
  It is the subarea of the plane that is under the point $(0/1)$ to the right, without the y-axis.
  
  Is this description enough or can we say also something else? (Wondering) The graphical representation is:
  
  View attachment 8510
  2. $C:=\mathbb{N}\times \{x\in \mathbb{R}\mid 0\leq x\leq 2\}$
  
  $C$ is the cartesian product of the natural number and the interval $[0,2]$. It s the set of points with two coordinates $(n,x)$, where the first coordinate is a natural number and the second coordinate is a real number in the interval $[0,2]$.
  
  What else can we say here? How does the graphical representation look like?
  
  (Wondering)
  
  $$ $$
  
• I want to give also the corresponding set fo the following:
  
  View attachment 8511
  
  It is $\{(x,y)\mid 0\leq x \leq 2, 1\leq y\leq 3\}$, right? Is this enough, or could we also justify that it is like that? (Wondering)
  
  $$ $$
  
• $F$ is the set of all points in the plane that are at least as far from the origin as $ P = (3 \mid 0) $. Does this mean that we have the set $$F=\{(x,y)\mid x^2+y^2\leq (x-3)^2+y^2\}$$ or have I understood wrong the definition of $F$ ? (Wondering)

 

[I like Serena]

Hey!

• I want to describe in words the following sets:
  
  1. $A:=\{(x,y)\in \mathbb{R}^2\mid x>0, y\leq 1\}$
  
  $A$ is the set of all pointgs where the first coordinate is positiv and the second one is less or equal to $1$.
  
  It is the subarea of the plane that is under the point $(0/1)$ to the right, without the y-axis.
  
  Is this description enough or can we say also something else? (Wondering) The graphical representation is:

Hey mathmari!

Seems fine to me. (Nod)
Btw, shouldn't it be $(0,1)$ instead of $(0/1)$?

• 2. $C:=\mathbb{N}\times \{x\in \mathbb{R}\mid 0\leq x\leq 2\}$
  
  $C$ is the cartesian product of the natural number and the interval $[0,2]$. It s the set of points with two coordinates $(n,x)$, where the first coordinate is a natural number and the second coordinate is a real number in the interval $[0,2]$.
  
  What else can we say here? How does the graphical representation look like?

It's a set of vertical line segments isn't it? (Thinking)

• $$ $$
  
• I want to give also the corresponding set fo the following:
  
  It is $\{(x,y)\mid 0\leq x \leq 2, 1\leq y\leq 3\}$, right? Is this enough, or could we also justify that it is like that?

It seems as if it should be $1 < y < 3$, shouldn't it?

• $F$ is the set of all points in the plane that are at least as far from the origin as $ P = (3 \mid 0) $. Does this mean that we have the set $$F=\{(x,y)\mid x^2+y^2\leq (x-3)^2+y^2\}$$ or have I understood wrong the definition of $F$ ?

What is $ P = (3 \mid 0) $? (Wondering)

 

[mathmari]

Seems fine to me. (Nod)
Btw, shouldn't it be $(0,1)$ instead of $(0/1)$?

Oh yes (Wasntme)

It's a set of vertical line segments isn't it? (Thinking)

So, we get the following, or not? (Wondering)

View attachment 8512

What is $ P = (3 \mid 0) $? (Wondering)

Oh I meant $P(3, 0)$. (Blush)

 

[I like Serena]

So, we get the following, or not?

Yes, although I'd be inclined to draw them vertically.
That is because we usually draw the first coordinate along the x-axis, and the second along the y-axis. (Nerd)

Oh I meant $P(3, 0)$.

Okay, so we're talking about the points that have at least the same distance to the origin as $(3,0)$ yes?
Its distance is $3$.
So we're looking at all points with $\sqrt{x^2+y^2} \ge 3$, don't we? (Thinking).

 

[mathmari]

Yes, although I'd be inclined to draw them vertically.
That is because we usually draw the first coordinate along the x-axis, and the second along the y-axis. (Nerd)

Oh yes, you're right! So, we get the following, don't we?

View attachment 8513

(Wondering)

Okay, so we're talking about the points that have at least the same distance to the origin as $(3,0)$ yes?
Its distance is $3$.
So we're looking at all points with $\sqrt{x^2+y^2} \ge 3$, don't we? (Thinking).

Ahh ok! So we have the set $$F=\{(x,y)\mid \sqrt{x^2+y^2} \ge 3\}$$ right? (Wondering)

 

[I like Serena]

Oh yes, you're right! So, we get the following, don't we?

Ahh ok! So we have the set $$F=\{(x,y)\mid \sqrt{x^2+y^2} \ge 3\}$$ right?

Yep. (Nod)