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Bachelier
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The function
##f(x) = \frac{x}{x+1} ##
is not a Homomorphism because f(1) ≠ 1..Am I correct?
##f(x) = \frac{x}{x+1} ##
is not a Homomorphism because f(1) ≠ 1..Am I correct?
Simon Bridge said:List the things f has to do to be a homomorphism.
Bachelier said:Assume we are in ℝ then
f(ab) = f(a).f(b) under multiplication
f(a+b) = f(a)+f(b) under addition
and that's all I find in my book. but I know we need to check some extra stuff.
micromass said:Could you please always list what structure you are working with. Saying that "f is a homomorphism" is a meaningless statement. You should state "f is a homomorphism of groups/rings/fields/algebras/lattices/..."
Also, be sure to always give the domain and codomain.
Simon Bridge said:You could also consider what sort of transformation is represented by f(x) ... i.e. is f(x) defined for all real x? Does it have to be if it is to be a homomorphism?
Bachelier said:Homomorphism of groups. mainly define our f: [0,∞) → ℝ.
That aside, a Homomorphism of groups must send the identity element of the domain to the identity element of the codomain, right?
Bachelier said:If I define my function f from ℝ to ℝ then the function is not defined at x = -1.
I take it from your question that a Homomorphism of groups must be well-defined on all elements of the group?
micromass said:OK, but [itex][0,\infty)[/itex] is not a group. So you can't talk about homomorphism of groups. Furthermore, a group only has one operation. So, saying that a homomorphisms of groups satisfy
[tex]f(x+y)=f(x)+f(y)~\text{and}~f(xy)=f(x)f(y)[/tex]
is not correct. Why not? Because now you're talking about two operations: addition and multiplication. A group is a set with only one operation (which satisfies some conditions.
So if you have a function [itex]f:(\mathbb{R},+)\rightarrow (\mathbb{R},+)[/itex] (I usually denote a group by [itex](G,*)[/itex], where G is a set and * is an operation on the set), then this is a homomorphism if and only if [itex]f(x+y)=f(x)+f(y)[/itex]. The multiplication has nothing to do with this.
In general, a function [itex]f:(G,*)\rightarrow (H,\oplus)[/itex] must satisfy [itex]f(x*y)=f(x)\oplus f(y)[/itex]. Nothing more.
If you want to talk about two operations (like addition and multiplication on [itex]\mathbb{R}[/itex]), then you have to talk about rings.
Yes. But what you mean with identity element depends on the group operation. In the group [itex](\mathbb{R},+)[/itex], the identity is 0. In the groups [itex](\mathbb{R}\setminus\{0\},\cdot)[/itex], the identity is 1.
Bachelier said:Now ##((0,∞), *)## where * is the regular multiplication is a group under ##*##, right?
micromass said:Yes, it is. You may be surprised to learn that [itex]((0,+\infty),\cdot)[/itex] is actually isomorphic (as group) to [itex](\mathbb{R},+)[/itex].
Bachelier said:After learning a long time ago that ##(0,1) \cong \mathbb{R}##, nothing surprises me anymore... lol :)
A homomorphism is a mathematical function that preserves the structure of a mathematical object. In simpler terms, it is a function that maps one algebraic structure to another in a way that respects their operations and properties.
For a function to be a homomorphism, it must preserve the operations and properties of the mathematical objects it is mapping. This means that the function must produce the same result as the original objects when the operations are performed on them.
To be a homomorphism, the function f must satisfy the following conditions:
While both homomorphisms and isomorphisms preserve the structure of mathematical objects, the main difference is that isomorphisms are bijective, meaning they have a one-to-one correspondence between the elements of the domain and codomain. Homomorphisms, on the other hand, do not necessarily have a one-to-one correspondence between the elements.
Homomorphisms are used in various fields of science, such as physics, chemistry, and computer science. They provide a way to study and understand the relationships between different mathematical objects and their structures. In physics, homomorphisms are used to describe the symmetries of physical systems, while in chemistry, they help in understanding the properties of molecules. In computer science, homomorphisms are used in data encryption and coding theory.