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bentley4
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I am unsure what is meant by binary operation.
Is the binary operation of addition : addition and subtraction or is it just addition?
Is the binary operation of addition : addition and subtraction or is it just addition?
bentley4 said:And the definition of ring acc. to wiki is: "An algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition."
So if we have the binary operation of just addition, we can do addition and subtraction since int. numbers can also be negative. But we also use devision in integer calculation next to multiplication so this is a third distinct binary operation. Yet it says a ring exists only of 2 binary operations, I don't understand.
Also, how can we prove the associativity of a ring if we only have 2 inputs? Or for what would associativity ever be useful if we can only use 2 inputs?
So when I calculate in Z I can only restrict myself to 2 inputs??
tiny-tim said:hi bentley4!
Aha, ok.tiny-tim said:there are infinitely many binary operations on a ring (a + 2b, a + 3b, etc) …
Don't we need a proof to make this kind of extrapolation?(or it is just unprovable and we assume it is right?)tiny-tim said:but two of them are defined as being the addition and multiplication in that particular ring
So a unitary operation is just a number then?tiny-tim said:a(b + c) and ab + ac are ternary operations (3 inputs) …
Where is this defined that you can do that, is it axiomised?tiny-tim said:we can have operations with as many inputs as we want
No.bentley4 said:So a unitary operation is just a number then?
The existence of Cartesian products is guaranteed by the axioms of set theory, and given any two sets X and Y, the set of functions from X into Y is also guaranteed to exist by the axioms of set theory. This implies that the existence of n-ary operations for any n is also guaranteed to exist.bentley4 said:Where is this defined that you can do that, is it axiomised?
You seem to be asking for a proof of the fact that the term "addition" is used only for one binary operation on the set. Obviously, that isn't something that needs to be proved. If you meant to ask something else, you will have to clarify.bentley4 said:Don't we need a proof to make this kind of extrapolation?(or it is just unprovable and we assume it is right?)
Here you seem to be asking why 1/n isn't always an integer.bentley4 said:Ok, but why isn't division defined for a binary operation in the ring of Z?
bentley4 said:Don't we need a proof to make this kind of extrapolation?(or it is just unprovable and we assume it is right?)
Ok, but why isn't division defined for a binary operation in the ring of Z? We need to define it if we want to use devision right? Or does the ring of Z not completely encompass Z as we normally use(such as in informatics)?
So a unitary operation is just a number then?
Where is this defined that you can do that, is it axiomised?
I still don't quite understand a unary operation.Fredrik said:No.
A unary operation on a set X is a function from X into X. (The word "unitary" means something else entirely).
A binary operation on a set X is a function from X×X into X.
...
An n-ary operation on a set X is a function from X×...×X (that's n copies of X) into X.
Ah. I think I understand now. I just meant to say in just R^1 for example, we can do a multiplication. For this multiplication, no cartesian product needs to be done.Fredrik said:The existence of Cartesian products is guaranteed by the axioms of set theory, and given any two sets X and Y, the set of functions from X into Y is also guaranteed to exist by the axioms of set theory. This implies that the existence of n-ary operations for any n is also guaranteed to exist.
So, for division, Z is not closed and isn't a ring(for division)?Fredrik said:Here you seem to be asking why 1/n isn't always an integer.
bentley4 said:Hi Fredrik, thnx for your response again.
Ah. I think I understand now. I just meant to say in just R^1 for example, we can do a multiplication. For this multiplication, no cartesian product needs to be done.
Multiplication can be derived from addition. such as 4*2= 4+4. But is addition defined? The fact that we say that 3 > 2 is due to language not set in the ZFC. I guess this is just accepted to be trivial. Just want to cover my grounds so I don't have to think of it again.
It seems that the concept you're having difficulties with is "function". Loosely speaking, a function from X into Y is an assignment of a member of Y to each member of X. (This isn't the definition of "function". It's just a useful way to think about functions). If f is a function from X into Y, then for each x in X, the member of Y that f associates with x is denoted by f(x). (X is called the domain of f. Y is called the codomain of f. The statement "f is a function from X into Y" is written as f:X→Y).bentley4 said:I still don't quite understand a unary operation.
Suppose we have a set L={2,3}
Then a binary operation on L is a function of LxL={(2,2),(2,3),(3,2),(3,3)} into L={2,3}.
What do you mean with into?
I don't understand what you're saying here. To specify a binary operation on L, let's call it b, you have to specify b(x,y) for each ordered pair (x,y) in L×L. Example: b(2,2)=2, b(2,3)=3, b(3,2)=3, b(3,3)=2.bentley4 said:From what I understand a binary operation would be (3,3)+(into?)(2,3)
This would equal (5,6), but this is not an element of any of these two sets.(only true in every case for an infinite set right?)
The multiplication operation on ℝ is a binary operation on ℝ, i.e. a function from ℝ×ℝ into ℝ.bentley4 said:I just meant to say in just R^1 for example, we can do a multiplication. For this multiplication, no cartesian product needs to be done.
There are many ways to define the integers. One way to do it is to say that if (Z,+,·) is a ring such that the set Z and the binary operations + and · satisfy certain conditions, the members of Z are called integers. Another way is to explicitly show that the ZFC axioms implies that such a ring exists. If you choose the latter option, the addition operation must be defined.bentley4 said:But is addition defined?
Right, if you view Z as a subset of the real numbers, it makes sense to say that it's not closed under division. If you think of Z as just a ring, division is simply undefined. It's still a ring, but not a division ring.bentley4 said:So, for division, Z is not closed and isn't a ring(for division)?
bentley4 said:And the definition of ring acc. to wiki is: "An algebraic structure consisting of a set together with two binary operations usually called addition and multiplication.
Suppose again the set L={2,3}. For u:L→L defined by u(2)=3 and u(3)=2 then u is a unary relation(operation).Fredrik said:A unary operation on your set L is a function u from L into L. So for each x in L, u(x) is in L.
Examples: The function that takes each non-zero real number x to 1/x. This is a unary operation on ℝ-{0}. If you need to see a unary operation on L, consider e.g. the function u:L→L defined by u(2)=3 and u(3)=2.
So an example of b:LxL→L is b(x)=2 How can a relation of pairs give a number? Can you give an example in this set?Fredrik said:I don't understand what you're saying here. To specify a binary operation on L, let's call it b, you have to specify b(x,y) for each ordered pair (x,y) in L×L. Example: b(2,2)=2, b(2,3)=3, b(3,2)=3, b(3,3)=2.
But a multiplication could just as well be a ternary operation right? If not, which are these operations called then?Fredrik said:The multiplication operation on ℝ is a binary operation on ℝ, i.e. a function from ℝ×ℝ into ℝ.
Ok, thnx.Fredrik said:There are many ways to define the integers. One way to do it is to say that if (Z,+,·) is a ring such that the set Z and the binary operations + and · satisfy certain conditions, the members of Z are called integers. Another way is to explicitly show that the ZFC axioms implies that such a ring exists. If you choose the latter option, the addition operation must be defined.
What do you mean? I just told you that this u is a unary operation on L. Is the word "other" missing from your question? As in "there doesn't exist any other unary operations on L, right?". In that case, the answer is that there are exactly four unary operations on L. Can you find them on your own?bentley4 said:Suppose again the set L={2,3}. For u:L→L defined by u(2)=3 and u(3)=2 then u is a unary relation(operation).
In this case there doesn't exist any unary operation u right?
If you mean "b(x)=2 for all x in L×L", then yes, this function b would be a binary operation on L.bentley4 said:So an example of b:LxL→L is b(x)=2
I don't understand what you're asking.bentley4 said:How can a relation of pairs give a number? Can you give an example in this set?
An example of b:L→LxL is b(x)=(2,3) How can a relation of numbers give an ordered pair as a result?
The definition of the real numbers includes two binary relations, one of them called "multiplication". You can of course use it to define a ternary relation, e.g. t(x,y,z)=x(yz).bentley4 said:But a multiplication could just as well be a ternary operation right? If not, which are these operations called then?
It depends on what binary operations on you have in mind when you say "addition" and "multiplication". I don't even know if what you have in mind are binary operations on L or on S.bentley4 said:For the case of S=LxL={(2,2),(2,3),(3,2),(3,3)}.
This set has no identity element for addition or multiplication.
Is not closed for addition and multiplication.
Is associative for addition and multiplication.
is commutative for addition and multiplication.
Is this correct?
Binary operations are mathematical operations that involve two operands or numbers. In other words, they are operations that require two inputs to produce a single output.
Addition in binary operations is the process of combining two numbers to produce a sum. In binary, addition involves carrying over a 1 when the sum of two digits is greater than 1.
Subtraction in binary operations is the process of finding the difference between two numbers. In binary, subtraction involves borrowing a 1 when the minuend is smaller than the subtrahend.
To perform addition in binary, you add each digit starting from the rightmost column, carrying over any 1s when the sum is greater than 1. To perform subtraction in binary, you borrow a 1 from the next column when necessary and subtract each digit starting from the rightmost column.
Binary operations are essential in computer science as all data in computers is represented in binary form, using only 0s and 1s. Addition and subtraction are fundamental operations in binary, and they are used in various computer algorithms and calculations.