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/sci/ - Science & Math

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2185834 No.2185834 [Reply] [Original]

i'm working on set theory, and I have a noob question.

why is {3}, the set with only one element, 3, different than 3? Is it because the set itself is an object?

>> No.2185837

3 is not a set. You can't blend apples and oranges.

>> No.2185846

I realize that 3 isn't a set, but {3}, a finite set containing one element, 3, is a set. A set is an object that contains other distinct objects. I don't understand why {3} is not the same as 3.

>> No.2185850

>>2185846

The set contains objects; the number 3 is an object. 3 is the element in this case.

Like what the previous poster said, you're mixing apples and oranges. The cardinality of {3} is (this will BLOW YO' MIND) is actually 1.

Set contains elements. In this case, the element is 3. The set =/= element. Sets are only elements in power-sets which is a whole other type of dealio.

>> No.2185852

Because {3} is "The set that contains the number 3 as it's first and only element". While 3 is just 3.

Therefore, does: "The set that contains the number 3 as it's first and only element" = the number 3?

Well, I hope you can see that they are most definitely not equal to each other. However, the first element of {3} = 3.

>> No.2185859

>>2185850

so basically I should think of {3} as a 3 within a container, and 3 as a standalone number? I'll learn the really deep stuff later, I'm just working on getting the basics of set theory and logic this morning. 56 page chapter and stuff. :)

>> No.2185864

>>2185859

Pretty much, yes. The set is not an object; it is a container of objects. 3, in this case, is an object.

The set {3,2,1} has the elements 1, 2, 3. Sets can also be arranged in any order, btw.

{3,2,1} = {1,3,2} = {1,2,3} = {2,3,1} = {2,1,3} = {3,1,2} are all the same.

>> No.2185872

>>2185859

By the way, it's correct nomenclature to refer to 3 as an element. You can interchange it with object, but it's preferable to call it an element. I can also have a set filled with squares, stars, boxes, chairs, etc. A set doesn't just include numbers.

Set theory can get pretty abstract which is why people prefer 'element' or 'object.'

>> No.2185883
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2185883

>>2185872

I get most of the other basic stuff, the distinction between a set containing an element, and the element itself is just sort of confusing. I guess because I think of a set as a restriction, so if a set is restricted to one element, 3, I'm inclined to consider them to be the same thing.

>> No.2185887

>>2185883
It can get weirder. A topology, for example, is a set of sets with certain properties such that it can contain the empty set. for example, {{a},{b},{empty set},{a and b}}

>> No.2185889

>>2185883
You should watch the MIT Lisp videos

It's like set theory in motion

>> No.2185891

>>2185883

Can it be that a white horse is not a horse?
Advocate: It can.
Objector: How?
Advocate: "Horse" is that by means of which one names the shape. "White" is that by means of which one names the color. What names the color is not what names the shape. Hence, I say that a white horse is not a horse.
Objector: If there are white horses, one cannot say that there are no horses. If one cannot say that there are no horses, doesn't that mean that there are horses? For there to be white horses is for there to be horses. How could it be that the white ones are not horses?
Advocate: If one wants a horse, that extends to a yellow or black horse. But if one wants a white horse, that does not extend to a yellow or black horse. Suppose that a white horse were a horse. Then what one wants [in the two cases] would be the same. If what one wants were the same, then a white [horse] would not differ from a horse. If what one wants does not differ, then how is it that a yellow or black horse is sometimes acceptable and sometimes unacceptable? It is clear that acceptable and unacceptable are mutually contrary. Hence, yellow and black horses are the same [in that, if there are yellow or black horses], one can respond that there are horses, but one cannot respond that there are white horses. Thus, it is evident that a white horse is not a horse

>> No.2185894

>>2185891
That arguments more relative in an old Chinese language.

Something about they didn't have clear grammatical constructs for talking about something as belonging to a set but not being the set.

>> No.2185904
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2185904

>>2185891

{horse} would be the set of all horses

{(color) horse} would be the subsets, separated by color, with it being evident that some of the sets would be empty

>> No.2185907

>>2185883

ur idiot professor introduced sets to you as Z, N, etc?

lulz. fuckin hate math profs that dont get math

>> No.2185918

3 is different to {3} in the same way an apple is different to a carrier bag with an apple inside

>> No.2185935

>>2185918
{3} is different to 3 as the cons'd pair of 3 and nil is from 3.

>> No.2185976

>>2185834
CAN YOU HELP FOR THESE PROBLEM I DONT HOW CAN I SOLVE THESE PROBLEM