In the sentence below what is meant by binary oreration. Does it mean “inner product”? And if so why don’t they just say that?
This means that the nonempty set of Lorentz transformations constitutes a
closed algebraic structure with a binary operation which is associative.
A binary operation on a set S is a function from the Cartesian product S[sup]2[/sup] to S. For example addition and multiplication are both binary operations on the set of real numbers. A binary operation * is associative if (ab)c = a(bc) for every a, b and c . When this is the case we can speak of “the product abc” without having to specify a bracketing, and the same holds true for larger products. By contrast, for example ( a - b) - c and a - ( b - c) are different things. Thus associative products are easier to handle.
When we are considering a set of functions, the usual binary operation is composition of functions ( which is associative on any set of functions). Your sentence is then saying that the composite of two Lorentz transformations is again a Lorentz transformation.
If V is a vector space an inner product is a map from V[sup]2[/sup] to C, which is quite different.
Pardon me? Everytime I think I know a little about math someone comes along and convinces me otherwise. Would it be possible to dumb that down a little? Why do they call it a binary operation? Whenever I hear the word “binary” I think it must have something to do with 2
It’s called a binary operation because it involves two operands. “Dumbing down” Jabba’s first sentence gives something like “a binary operation is a rule which produces a member of a specific set of from an ordered pair of elements of the set”.
Given a set S, the Cartesian product is the set of all ordered pairs (a,b), where both a and b are in S. A binary product takes such an ordered pair and maps it onto a single element of the original set S. The way to think about this is as a product which maps the pair (a,b) onto the product a*b, where we use * to denote an arbitrary binary operation. The idea is that we can take any two elements of the original set and combine them to form a third element of S. This is the point of the “binary”: we combine two elements of S at a time.
Examples:
Addition is a binary product on R, the set of real numbers: given any two real numbers a and b, we can form their sum a+b. In terms of the formal notation, the operation maps the ordered pair (a,b) onto the single number a+b.
Multiplication is a binary operation on R: given any two real numbers a and b, we can form their product ab.
Addition and multiplication are also binary products on C, the set of complex numbers, Z, the set of integers, Q, the set of rational numbers, and N, the set of natural numbers.
Division is not a binary operation on R: division by 0 is not defined. This illustates the first point you have to check for a binary operation: it must be possible to combine any two elements of S. Division is a binary operation on the set of non-zero real numbers and the set of positive real numbers.
Subtraction is not a binary operation on the set of positive integers: if a and b are positive integers, it is not necessarily true that a - b is a positive integer. However, subtraction is a binary operation on each of C, R, Q and Z. This illustrates another of the key points you must check for a proposed binary operation: the “product” of two elements of S must again be an element of S.
Matrix multiplication is a binary operation on the set of all 2-by-2 matrices ( also on the set of all 3-by-3 matrices,…)
Let S denote the set of all functions from R to R, that is the set of all functions which map real numbers onto real numbers. We write mappings on the write, so xf is the image of the point x under the function f. Given two functions f, g in S, we define their product fg to be the composite function, defined by
xfg = (xf)g
i.e, to find the image of any point x under f*g, first apply f to x, to obtain xf, and then apply g to xf to get (xf)g.
Then * is a binary product on S. Sets of functions under composition are of the highest importance in algebra.
Let me know how much sense all that makes and we can move on from their.
It made a lot sense, and I really appreciate your answer.
I’ve self-studied calculus, ordinary and partial differential equation, and a real simple linear algebra text (Elementary Linear Algebra by Anton Rorres). But I haven’t run across this terminology before, is this covered in Real Analysis or something?
In the sentence I quoted above it says a closed algebraic structure with “a” binary operation. Why the “a”? Does this refer to a “composition of two functions” or just any old binary operation?
Why are Sets of functions under compositionof the highest importance in algebra.? Does any of this stuff have to do with vector spaces and the closure axiom? Thanks again for your help, and just ignore me if I ask too many stupid questions. The subtle deeper meanings of absract math tends to ellude me.
:smack: Move on from their what? I can’t believe I wrote that. One day I’ll make a post that makes me look intelligent rather than illiterate.
Firstly, don’t worry, your questions aren’t stupid. These are difficult subjects. Just keep plugging away and keep asking questions when you get stuck and we’ll get there/their/they’re in the end.
Now let’s recap associativity. A product * is associative if, for every choice of a, b and c,
(ab)c = a(bc)
This means that to find the product of a, b and c ( in that order), we have two choices:
(i) Find the product ab. Call it d. The find the product dc.
(ii) Find the product bc. Call it e. Then find the product ae.
The associative law says that these two products will be equal.
To relate this to the earlier examples:
1/2/3) Addition and multiplication are associative.
4) Not associative: (a/b)/c is in general not equal to a/(b/c)
5)Not associative: (a - b) - c is not equal to a - ( b - c) in general.
6) Associative
7) Associative
A product * is called commutative if, for every a and b,
ab = ba
Associativity and commutativity are quite independent. In example 6, matrix multiplication is associative but not commutative. You cannot change the order of terms in a product unless * is commutative.
The operation of taking the composite of functions is associative. When we consider a set of functions it is almost always composition that we have in mind for the product. When using this as our product, we must take particular care to check that closure holds. So the force of the sentence you quoted in your OP is that the composition of two Lorentz transformations is again a Lorentz transformation. I would assume that this was proved just before the part you quoted, and that this is the “This” at the beginning of the quotation.
For the use of the word “a”, I would read the sentence as:
closed algebraic structure with ( a binary operation which is associative)
The “closed” is just the point I made in example 5 of my second post. The product of two elements of S again lies in S. The sentence is stressing that the product is associative.
Finally, you asked why sets of transformations are important. This comes from group theory. There is a result, called Cayley’s Theorem, which says that every group is isomorphic to a group of transformations ( that is 1-to-1 correspondences on a set).