what is a group compared to a set? i get that a set is a collection of objects, be it numbers or functions, that have constraints on the domain or range, but what does a group add?
better yet, how does one extend from a set to a ring and field? from what i understand, rings allow you to add and subtract subsets, and fields allow you to multiply and divide? so i ask, where does a group sit in this framework?
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Literally Google the definition of 'group', 'ring', and 'field', you poor, helpless retard
ok,A Set (mathematics) is essentially a collection of any sort of objects .These objects are called the elements or the members of the set . ... A Group (mathematics) is an algebraic structure and a set closed under certain specific operations .
but a field already fits under this criteria so how is a group any different? i bet you don't even know you brainlet.
Groups are algebraic structures on sets closed under one operation. Rings already have two operations. Fields are rings with some more stuff (a multiplicative identity, for instance).
Look up the axioms of each structures to see the difference. Groups are create tools to talk about symmetry, be it geometric or physical. Rings generalize properties of the integers to polynomials, matrices and other objects. Commutative rings in particular form the basis of a field of mathematics called (modern) algebraic geometry. Fields are useful in general, you can define vector spaces on them and do calculus, provide you add some more structure on them (something called a topology). Finite fields are used in encryption, too.
so a group must have one operation whereas rings and fields have more than 2?
Every field is also a group, a field is a more restrictive definition.
Groups are required to have one operation that does certain things.
Rings and fields are required to have two operations that do certain things (fields have more requirements).
i don't understand. if a group must have one operation but a ring and field must have two, how can every field be a group as well?
A set with one associative, inversible operation is a group.
A set with two associative, inversible operations, one of them being commutative, and obeying distributivity, is a ring.
A ring has two operations, so it has one operation.
A ring is a group.
I really fucked this up but you get the idea
You are sorta thinking about this wrong. We typically say that a collection of objects *and* an operation forms a group. For example, the reals form a group under addition.
We talk about a field as a collection of numbers and 2 operations. For example, the reals are a field under addition and multiplication.
Some sets can form different groups if you choose different operations. For example the integers are a group under addition. Also they are a group under multiplication. However, they are not a group under division.
If a set of numbers is a field under 2 operations, then it can be a group under each of those operations (I think, it's been a while since I saw the definition of a field).
Oh and a similar thing applies to rings like
says. A field is in fact a ring with more restrictions (once again I'm pretty sure this is true, but I might be missing a small detail).
Rings and fields are groups under their first operation. This is inherent in the definition.
This is really the wrong way to learn this stuff... pick up an intro text on Modern Algebra. That will answer all your questions and more.
is the Springer book on Algebra the standard read? i saw that my prof had one.
There are many undergraduate standards: Herstein, Artin, Dummit&Foote, Pinter, Saracino... Just pick the one you like best, for me it was a mix between Herstein and Artin.
not sure why people aren't communicating it to you clearly, make an analogy to shapes for instance:
What is the condition for a shape to be a square? All sides (4 of them) are equal
What is the condition for a shape to be a rectangle? Opposite sides (4 of them) are equal.
Now, is it not true that: all squares ARE rectangles, but all rectangles ARE NOT squares? Squares fit under the condition for a rectangle and so are called rectangles too.
In the same way, a field is a ring in that it has more restrictive properties (existence of the multiplicative inverse), analogously to our case of geometry: all fields ARE rings, but all rings ARE NOT fields.
Same thing with rings vs groups. All rings ARE groups (they satisfy condition for a group), but not all groups ARE rings.
How about a nice Venn diagram that explains this better?
This is correct
>Same thing with rings vs groups. All rings ARE groups (they satisfy condition for a group), but not all groups ARE rings.
thats just confusing.
a group is a set with ONE law of composition thats associative, has a neutral element and inverses
a ring has TWO operations, addition and multiplication, where addition is ALWAYS a commutative group and addition distributes over multiplication
a field is a ring such that all non zero elements form a commutative group under multiplication. so a field has two groups with distributive law.
but honestly learning from a self-contained algebra book is the best
{..., -2, -1, 0, 1, 2, 3, .... } = Z - a set
- a group
- a ring
- a field
Read the god damn definitions
Also
- a ring, but not a field
what's R
2. You will immediately cease and not continue to access the site if you are under the age of 18.
I just asked what R is, I'm 20 but not a maths student, you prick
But I saw a few lectures by a maths phd who says the real numbers don't exist. I was under the impression we were talking about actual maths, not your fake "I believe in God because I do"-tier extension of the rational numbers, and that's what I wanted explained to me. Apparently you're just another one of those infinitist idiots who doesn't have any clue about what they're talking about and then tries to shame other people for not following the same REALigious dogma that he does.
Real numbers do exist, I'm cramming these theorems right now to prepare for exams.
All you need to do to prove that is
1) define what is R
R is an ordered field, characterized by the continuity axiom (we, Russians, call it "the continuity axiom", don't know about other languages), that is, given 2 subsets of R - A and B. If for any a from A and b from B, a
...
I took it out of boredom
if the real numbers exist, then can you please enumerate them for me? I just need the bijection between them and the naturals, that's all
> all infinite sets are countable
Low quality b8 m8, did you even read my post?
>the integers are a group under multiplication
How can something be so wrong?
Who said that?
...
Ayy
The dude needs to read some algebra asap
Wilderberger pls