Geometric Point of View of sets

[DumpmeAdrenaline]

A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't.
The geometric point of view of sets- a set can be viewed as being contained with a closed curve.
1) How does the geometric view of set show the common property among the elements of a set? We can list the members of the set inside a closed curve call it some upper case letter but still the picture doesn't highlight the common definition or property the members share.

My thoughts: We pick any arbitrary region in space, R. In order for an object to be an element ( reside inside R) it has to obey certain rules, in the same way an individual is subject to certain laws of a legal system which must be respected in order to be part of it.
Can we view a set as some in region in space+ preset conditions?
Expanding further, can subsets be viewed as a region within the region of the original set ( conditions of R) + Additional conditions?

 

[Office_Shredder]

A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't.

Is this your definition of a set, or just a convenient way to describe most sets you encounter in the real world?

What are the conditions you would test an object on to decide if it's in the set of natural numbers?

 

[DumpmeAdrenaline]

Is this your definition of a set, or just a convenient way to describe most sets you encounter in the real world?

What are the conditions you would test an object on to decide if it's in the set of natural numbers?

Natural Numbers are whole numbers (not decimals nor fractions) used for counting as in 0,1,2... etc

 

[Office_Shredder]

What are whole numbers? If you say something involving real numbers or rational numbers, then please define those next. Hopefully you realize at some point there's no great description of what the set actually contains. And in fact you've gone backwards, normal set theory usually starts by constructing the natural numbers, and then defining integers and rationals etc from there.

My point here is that a set is simply a collection of objects, and to test if something is an element of a set, you simply see if it's in the collection. It turns out that a very normal way to construct sets in practice is to start with one set $A$, and from there say $B\subset A$ is the set of all things in $A$ satisfying some condition, but it's not actually necessary to define a set like this, you can construct sets in other ways (in particular, this doesn't tell you how to make the original $A$ set to begin with).

Here's an example that gets to your original question. Suppose I want to describe the graph of a function. I could describe it as $\{(x,y)\in \mathbb{R}^2\ :\ y=f(x)\}$, or I could just describe it as the collection of all points of the form $(x,f(x))$ where $x\in \mathbb{R}$. From a set theoretic point of view, the former is safer (in most versions of set theory there's an axiom that describing subsets using conditions is guaranteed to construct a valid set), but the latter is in fact a description of a valid set, and you could imagine accepting its existence independent of knowing that $\mathbb{R}^2$ even exists.

In the extreme case you could imagine me just picking a literally arbitrary collection of points of the form $(x,y)$, and there might be no way to test if a point in $\mathbb{R}^2$ is an element of the set other than you just look at the whole collection and see if it's in there.

 

[DumpmeAdrenaline]

By my definition I am stuck in the loop of finding a definition for a set each time instead we look for a starting point, a set which exists on its own independent of any definitions or rules from we can form new sub-collections from the original set based on some conditions we apply.

Applying what I have learned in your example, what we wish to study is the set of all points in the x-y plane and we impose a condition y=f(x) namely f maps x into y to form a subset (x,f(x))
The latter (x,y) doesn't emphasize f. So the only we can tell if (x,y) belongs or not is to list all members and match.

 

[Mark44]

Applying what I have learned in your example, what we wish to study is the set of all points in the x-y plane and we impose a condition y=f(x) namely f maps x into y to form a subset (x,f(x))
The latter (x,y) doesn't emphasize f. So the only we can tell if (x,y) belongs or not is to list all members and match.

Do you think you can list all members of the set $\{(x,y)\in \mathbb{R}^2\ :\ y=3x \}$? Take your time ...

 

[Mark44]

The geometric point of view of sets- a set can be viewed as being contained with a closed curve.

Although Venn diagrams do something like this, the closed curve acts merely as a container for the objects in the set. Other than this, I don't see much point in taking a geometric view of sets.

 

[PeroK]

The definition of a circle or a sphere is the set of all points the same distance from a fixed central point.

Similarly, conic sections may be defined in terms of fixed points and distances.

Aren't those geometric definitions of sets?

 

[DumpmeAdrenaline]

Do you think you can list all members of the set $\{(x,y)\in \mathbb{R}^2\ :\ y=3x \}$? Take your time ...

No, I cannot list an infinite set of elements. Resort to trial and error, plug in x and if it yields the same y as in y=f(x)=3x then x belongs to the set.

 

[DumpmeAdrenaline]

Empty set- A set whose test for membership is so stringent that no object can survive it. For example, find a number greater than 5 and less than 3. Definitely an objective and a well defined test but no number satisfies both conditions. Coming to what is bothering me, how is an empty set a subset of any considered set? By definition of a subset, all of its elements are also elements of the original set, but an empty set has no members? Doesn't the definition imply at least one member?

 

[PeroK]

By definition of a subset, all of its elements are also elements of the original set, but an empty set has no members? Doesn't the definition imply at least one member?

It doesn't. This is sometimes called a vacuously true condition. In any case $\emptyset \subseteq X$, where $X$ is any set.

 

[Mark44]

So the only we can tell if (x,y) belongs or not is to list all members and match.

No, I cannot list an infinite set of elements.

Then your technique of listing all members to see if you get a match doesn't work.

Similarly, conic sections may be defined in terms of fixed points and distances.
Aren't those geometric definitions of sets?

Yes, but sets such as these are already geometric objects. If the set elements don't represent some geometric object, I don't see how a geometric representation of the set makes any sense. For an example, how would this set be represented geometrically? The set of real numbers in [0, 1] such that the decimal representation has a '7' digit in the thousandths' place.

 

[PeroK]

For an example, how would this set be represented geometrically? The set of real numbers in [0, 1] such that the decimal representation has a '7' digit in the thousandths' place.

@zenterix and I were struggling with something like that recently:

https://www.physicsforums.com/threa...pivaks-calculus-ch-4-5-graphs-limits.1010265/

 

[Mark44]

@zenterix and I were struggling with something like that recently:

https://www.physicsforums.com/threa...pivaks-calculus-ch-4-5-graphs-limits.1010265/

Interesting. I didn't see that thread. The example I posted was just something I picked at random.