MHB Describing Sets: A Comprehensive Guide

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The discussion focuses on describing various mathematical sets in detail. Set A consists of points in the plane where the x-coordinate is positive and the y-coordinate is less than or equal to 1, forming a region to the right of the y-axis. Set C is the Cartesian product of natural numbers and the interval [0,2], represented as vertical line segments in a graph. The participants clarify the representation of another set defined by the coordinates (x,y) within specified bounds, suggesting it should be expressed with strict inequalities. Finally, they confirm the correct interpretation of set F, which includes all points at least three units away from the origin, expressed mathematically as F = {(x,y) | √(x² + y²) ≥ 3}.
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Hey! :o

  • 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)
 

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mathmari said:
Hey! :o

  • 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 $(0/1)$?

mathmari said:
  • 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)

mathmari said:
  • $$ $$
  • 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?
mathmari said:
  • $F$ 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)
 
Klaas van Aarsen said:
Seems fine to me. (Nod)
Btw, shouldn't it be $(0,1)$ instead of $(0/1)$?

Oh yes (Wasntme)
Klaas van Aarsen said:
It's a set of vertical line segments isn't it? (Thinking)

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

View attachment 8512
Klaas van Aarsen said:
What is $ P = (3 \mid 0) $? (Wondering)

Oh I meant $P(3, 0)$. (Blush)
 

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mathmari said:
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)

mathmari said:
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).
 
Klaas van Aarsen said:
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)
Klaas van Aarsen said:
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)
 

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mathmari said:
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)
 

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