What are subsets and proper subsets in set theory?

[madmax2006]

Why does "Set theory" exist?

Is there actually a use for Set theory? I understand why Algebra, Calculus, Trig, Physics, etc.. exist. Does anyone ever use it at their job (besides at some type of school).

 

[HallsofIvy]

Since I have worked in schools all my life, I can't answer that- but I have a question of my own: why do you think something must be used outside of school in order to be useful? A major reason for set theory is that it makes the terminology of other forms of mathematics easier. It makes it easier, while you are in school, to learn those things you will use outside of school.

Aside from that, of course everyone "uses" set theory everyday- even if they don't know it or have never learned set theory. Humans can't help but categorize things and put them into 'sets'- that's how we think. And that's what set theory is about.

 

[CRGreathouse]

Of course in computer programming, data structures are often based on set theoretic constructs -- a "bag" is just a multiset, for example.

 

[fourier jr]

i worked with a lady who was going to be a teacher & i was surprised that she knew the word "cardinality". she was doing developmental psych or something & that's where she learned it, and it had the same meaning in her psych course as it does in math.

 

[SW VandeCarr]

To answer your original question: "Why does Set Theory exist?"; Cantor's original motivation came from his work on Fourier series and transcendental numbers (see 2.Origins of Set Theory, second paragraph, in the linked article).

http://plato.stanford.edu/entries/set-theory/l

 

[Greg Bernhardt]

Isn't it useful for music?

 

[martharon]

Might be they are the only easy paper in maths, i would thank the person who founded it.

 

[Ben1220]

When I first read the question I thought you meant, "why did mathematicians create set theory the way that they did" And I would have answered "set theory exists because mathematicians say it does." Meaning that the axioms chosen are pretty arbitrary. But now reading the body of your post I think you mean, what are the applications of set theory. Apart from in computer science and computer programming, where it is actually used quite a lot, that's not really the point. Set theory is used by other areas of mathematics, it is the foundations on which everything is built. There is no royal road to learning maths. You want to know things properly, you need to delve down into set theory and mathematical logic, even if you're not actually going to go around explicitly unioning sets every day in the future.

 

[madmax2006]

When I first read the question I thought you meant, "why did mathematicians create set theory the way that they did" And I would have answered "set theory exists because mathematicians say it does." Meaning that the axioms chosen are pretty arbitrary. But now reading the body of your post I think you mean, what are the applications of set theory. Apart from in computer science and computer programming, where it is actually used quite a lot, that's not really the point. Set theory is used by other areas of mathematics, it is the foundations on which everything is built. There is no royal road to learning maths. You want to know things properly, you need to delve down into set theory and mathematical logic, even if you're not actually going to go around explicitly unioning sets every day in the future.

Indeed, I was wondering what the applications of set theory are. I had a talk with my Data Structures & Algorithms I professor & she showed me a few ways it helps. Thanks.

 

[JSuarez]

Set theory (or theories: there is more than one) has obvious uses in Mathematics and Philosophy, where it's the most developed theory about classes of objects that we have, and it's currently regarded as the most stable foundation for Mathematics. In that sense, people working in these fields "use" Set Theory, but your question seems to disqualify these as valid, or worth pursuing, uses. So, in what seems to me to be your sense of "use", the answer is no.

Your question is an old one: are purely theoretical enquiries worth pursuing? Some societies said "no", others "yes", still others choose some form of middle ground. In the end it's a matter of choice.

 

[raf1qu3]

hmmm, Tough question. Let me try my best to answer this. first off i am a computing student and have Logic as a core module... in semester 1 of year 1 we covered set theory. it didnt make much sense to me until i started to do apply this to real life situations...

for example.

lets take a set, What is a set? you may think it can only be numbers and letters at this stage BUT they can be substituted by anything, Suppose there are four objects: {ball, car, doll, gun} and four persons: {John, Mary, Ian, Venus}. Suppose that John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing. Then the binary relation "is owned by" is given as

R=({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of people, and the last element is a set of ordered pairs of the form (object, owner).

The pair (ball, John), denoted by ballRJohn means that the ball is owned by John.

Now how does this benefit me?

with this understanding from set theory, i can create two database one with the Object and one with the persons and know exactly how my relation tables should look like. When i need to get a perticular output from a program which utilizez these database, i won't have o make any changes to the database i will only need to input the function which i learned from set theory into my coding and only then will i get an accurate output. This is more obviouse when creating a calculator to perform a perticular funtion. Without basic knowledge of set theory, making advance programs will be very diffecult.

 

[Tac-Tics]

Set theory is grammar for all other realms of mathematics. You use it all the time in all other branches of math without ever realizing it. Learning about it often helps your clarity and articulation in math.

It's all about proofs. A set can be equal to another. A set can include another. Nothing more. It's very easy to write proofs about them. And *everything* is a set. You can define integers, real numbers, functions, ordered pairs, any kind of mathematical object of interest as a certain kind of set. For mathematicians wanting a rigorous framework, set theory is king.

But if you aren't interested in proofs (see: the rest of the world), set theory isn't immediately practical.

 

[madness]

http://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology#Molecular_set_theory

"Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in Mathematical Medicine.[26] Molecular set theory (MST) is a mathematical formulation of the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets. In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine."

 

[KarenLeigh]

What is the definition of subsets and proper subsets? I have asked that question and got no answers.

 

[SW VandeCarr]

What is the definition of subsets and proper subsets? I have asked that question and got no answers.

Every set is a subset of itself, but a proper subset must exclude at least one member of the containing set.

 

[HallsofIvy]

What is the definition of subsets and proper subsets? I have asked that question and got no answers.

Really? Who did you ask? Those are fairly fundamental definitions and usually in the first section of any text on sets.

A is a subset of B if and only if every member of A is also a member of B.

"Proper" subset is not quite as standard. A few texts use "proper" subset to mean any subset of B other than B itself. Most use it to mean any subset of B other than B itself or the empty set.