What is the difference between Field and Set ?

[weetabixharry]

What is the difference between "Field" and "Set"?

I was browsing through the Notation section of my favourite book and noticed that symbols were defined for the "sets" of real, integer and natural numbers... but for the "field" of complex numbers.

Is the terminology different because a complex number comprises two components (real and imaginary)?

I don't want a cripplingly in-depth definition - Wikipedia tells me a field is a "commutative ring whose nonzero elements form a group under multiplication". This means almost nothing to me.

 

[dodo]

I can try a "cripplingly simple" description of these terms -- very lacking in detail, but one that may allow for a broad, initial view.

You probably know what a "set" is: a collection of elements. If you enhance a set by providing it with an operation (like addition on the set of integers, for example), you obtain something with a bit more structure, called a "group". Add a second operation (like multiplication), and you get a more complex structure called a "ring". Consider now one such "ring" but with the ability to divide (you can't, in general, divide two integers and expect always to obtain another integer... but you can do that in the set of fractions, for example) and you have a "field". All fields are rings; all rings are groups, and all groups are sets.

The integers, with addition, form a group. The integers with addition a multiplication, a ring. The rationals (fractions), the reals, and the complex numbers, each constitute a field under addition and multiplication. These are the most direct examples -- but the terms "group", "ring" and "field" were created not as much to study the integers or the reals, but to extrapolate the basic, underlying machinery of integers and reals into other, more abstract sets and operations, and figure what is essential to all of them, what are the fundamentals that make them tick.

 

[I like Serena]

Hi weetabixharry!

A set only consists of elements.
A field is a set combined with addition and multiplication (that are properly defined according to a set of axioms).

So the real numbers also form a field.
And the complex numbers also form a set.

 

[Deveno]

I was browsing through the Notation section of my favourite book and noticed that symbols were defined for the "sets" of real, integer and natural numbers... but for the "field" of complex numbers.

Is the terminology different because a complex number comprises two components (real and imaginary)?

I don't want a cripplingly in-depth definition - Wikipedia tells me a field is a "commutative ring whose nonzero elements form a group under multiplication". This means almost nothing to me.

well, there are different kinds of "structures" we can realize on sets.

a set is just a collection of objects (naively speaking, i don't want to get into the complexities of Zermelo-Fraenkel set theory, the current favorite). the set itself has no structure, we can't "add" or "multiply" or "average" set elements, all we can do is say things like: in the set, not in the set.

(think about it: {Alice, Bob} is a perfectly good set, but what does Alice+Bob, even mean?).

but it is possible to create sets that "interact" with themsevles. the obvious example is ordinary natural numbers, where we can do arithmetic.

a field is a highly organized set, with two compatible structures: addition and multiplication. furthermore, these structures are "complete", in the sense that the two equations:

a + x = b
ax = b (if a is not 0)

always have a unique solution for x(in other words, we can do all of our usual algebraic manipulations in a field, they are good for solving equations in).

tl;dr version: all fields are sets, but not all sets are fields. fields have special properties.

 

[SteveL27]

I was browsing through the Notation section of my favourite book and noticed that symbols were defined for the "sets" of real, integer and natural numbers... but for the "field" of complex numbers.

Is the terminology different because a complex number comprises two components (real and imaginary)?

I don't want a cripplingly in-depth definition - Wikipedia tells me a field is a "commutative ring whose nonzero elements form a group under multiplication". This means almost nothing to me.

Ok here's a cripplingly easy explanation.

Say you had the familiar integers: 0, 1, 2, 3, 4, 5, ... and their negatives. But you forgot how to add or multiply. So you can recognize that, say, 47 is different than 169. You can identify each of the numbers. But you can't DO anything with them.

That's a set.

Now if someone hands you the addition and multiplication tables so that you can do arithmetic -- that's a commutative ring. In other words it's a set of objects along with some rules for doing operations on them.

If besides multiplying you can also divide, then its a field. For example you can't divide 3 by 4, so the natural numbers aren't a field.

But the complex numbers are a field. You can in fact divide one complex number by any other nonzero complex number and get a unique answer.

So the complex numbers, along with the usual addition and multiplication, are a field.

 

[Guffel]

Thanks guys (Dodo in particular) for the very nice introductions to abstract algebraic structures! I think many books on the subject would be significantly improved by including any or all of the above texts.

 

[weetabixharry]

Many thanks for these insightful responses. I've done a bit more reading and now seem, at least, to have some sort of a hand-waving grasp on the topic. This stuff is so profound, it's slightly terrifying. I'm worried that if I go scraping around so close to the truth underlying mathematics, I'll bump into god, or my head will burst into flames or something. I shall carry around a bucket of water, just in case.

 

[I like Serena]

I like the way how things have been abstracted into higher math concepts.

But to be realistic, none of it matters in real-life!
I don't know of any real-life applications other than that it helps to know how to generalize stuff and how to think logically without making assumptions!

 

[micromass]

I don't know of any real-life applications

There is a very large number of real-life applications! One application is in coding theory. We can talk about codes whose words are subsets of a finite field. So knowing about finite fields, and their properties certainly can help you in real life.

I admit, one does not NEED to know about fields to be able to do coding theory. But the theory is made substantially simpler when we do.

 

[I like Serena]

Hey micro, sorry about that!
I'm afraid my specialty is more in applied mathematics! :shy:
Isn't coding theory mainly about Euler's totient function, the Chinese remainder theorem and some such?
You don't need fields for that, do you?

 

[micromass]

Hey micro, sorry about that!
I'm afraid my specialty is more in applied mathematics! :shy:
Isn't coding theory mainly about Euler's totient function, the Chinese remainder theorem and some such?
You don't need fields for that, do you?

Coding theory is large, and you don't need fields everywhere.

Here is where fields come in handy: the problem is to discover error detecting and error correcting codes. So for example, the user must type in a 10 digit number (for example the ISBN number of a book). But if the user makes a small mistake, then the system must recognize this.

How do we accomplish that?? Well, we have 9 digits which can vary and we have one control bit which is basically the sum of the previous digits mod 11. For example, if the first 9 digits are 154324456, then the last digit must be

$$1+5+4+3+2+4+4+5+6=1 \ \ (mod~11)$$

So your code will be 1543244561.

This code can detect errors: if you type something wrong, then you can calculate things (mod 11) and you'll see that something went wrong.

You can also have error correcting codes: codes which don't only know that there has been a mistake, but can actually know what the correct code was!

In my previous example, you see that we already worked with the field with 11 elements. The word field wasn't really important here, but it is once you dig deeper. A lot of codes are so called linear codes. That means that the correct codes form a subspace of $\mathbb{F}_n$. We can now study this with linear algebra...

I hope this was useful

 

[Deveno]

Hey micro, sorry about that!
I'm afraid my specialty is more in applied mathematics! :shy:
Isn't coding theory mainly about Euler's totient function, the Chinese remainder theorem and some such?
You don't need fields for that, do you?

while this is true, what is often not brought up explicitly, is the role that prime numbers play.
in modular arithmetic, to be able to "undo" something you've done, you need to be working with invertible elements. if your modulus is not a prime (if you are just working mod n, for a composite number n), it is possible to encode information, and be unable to decode it (as the decodings are not unique).

RSA actually takes this idea and runs with it: it uses mod pq, which is composite, which results in exponents in U(pq) = U(p) x U(q). to crack it, you need to find p and q, while only knowing n (the definition φ(n) = φ(pq) = φ(p)φ(q) doesn't help you if you don't know p and q). if p and q are large primes, factoring pq is considerably harder than checking for primality.

the point being: if we didn't start with mod p and mod q, we would have uniqueness problems in devising a suitable encrypt/decrypt method. it's still possible, but the "keys" have to be chosen much more carefully, and the chinese remainder theorem can't be used to make the algorithm more efficient.

finite fields are also used in signal-processing equipment (to prevent signal degradation through transmission losses), using a similar approach to micromass's ISBN checksum example.

 

[SteveL27]

I like the way how things have been abstracted into higher math concepts.

But to be realistic, none of it matters in real-life!
I don't know of any real-life applications other than that it helps to know how to generalize stuff and how to think logically without making assumptions!

Computer security -- in particular, public key cryptography -- is based on modular arithmetic.