Describing Sets: A Comprehensive Guide

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In summary, the conversation discusses different sets and their descriptions. The first set, $A$, is a subarea of the plane where the first coordinate is positive and the second is less than or equal to 1. The second set, $C$, is the cartesian product of the natural numbers and the interval [0,2]. The third set is the corresponding set for a given set, and the fourth set, $F$, is the set of points that are at least as far from the origin as the point (3,0). The graphical representations for these sets are also discussed.
  • #1
mathmari
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Hey! :eek:

  • 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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  • #2
mathmari said:
Hey! :eek:

  • 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)
 
  • #3
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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  • #4
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).
 
  • #5
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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  • #6
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)
 

FAQ: Describing Sets: A Comprehensive Guide

1. What is the purpose of "Describing Sets: A Comprehensive Guide"?

The purpose of "Describing Sets: A Comprehensive Guide" is to provide a comprehensive and detailed explanation of how to effectively describe sets in mathematics and other scientific fields. It aims to clarify misconceptions and provide a clear understanding of the concept of sets and their properties.

2. What is a set and how is it different from a group or collection?

A set is a well-defined collection of distinct objects or elements. It is different from a group or collection in that the elements of a set are unordered and do not have any specific arrangement or structure. Additionally, a set cannot contain duplicate elements.

3. What are the different ways to describe a set?

There are several ways to describe a set, including listing the elements, using set-builder notation, and using verbal descriptions. Listing the elements involves listing all the objects or elements within a set, while set-builder notation involves using mathematical symbols and conditions to describe the elements of a set. Verbal descriptions use words and phrases to describe the elements of a set.

4. How do you determine the cardinality of a set?

The cardinality of a set is the number of elements in the set. To determine the cardinality of a set, you can count the number of elements in the set or use the concept of one-to-one correspondence, where each element in the set is paired with a unique natural number. This number corresponds to the cardinality of the set.

5. What are the common properties of sets and how are they used in describing sets?

The common properties of sets include intersection, union, and complement. Intersection is the set of elements that are common to two or more sets, while union is the set of all elements in the given sets. Complement is the set of elements that are not in the given set. These properties are used in set operations and can be used to describe sets using mathematical notation.

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