F[a][b] vs F[a,b]: What's the Difference?

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In summary, F[a][b] represents the set of polynomials in b with coefficients from the set F[a], while F[a,b] represents the set of bivariate polynomials in a and b with coefficients from the set F. The two notations may seem different, but they are actually isomorphic.
  • #1
futurebird
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What is F[a]?

I'm having a hard time with notation. If F[x] is the set of polynomials in x, then what is:

F[a]?

How is it different from F[a,b]?


And help would be great?
 
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  • #2


futurebird said:
I'm having a hard time with notation. If F[x] is the set of polynomials in x, then what is:

F[a]?

How is it different from F[a,b]? And help would be great?

The very pleasant thing is that F[a,b] really is isomorphic to F[a], just like we want it to be!
 
  • #3


Hurkyl said:
The very pleasant thing is that F[a,b] really is isomorphic to F[a], just like we want it to be!


Great! But what the heck are they? :tongue2:
 
  • #4


futurebird said:
Great! But what the heck are they? :tongue2:
F[a] is the set of polynomials in b whose coefficients are elements of F[a]
F[a,b] is the set of bivariate polynomials in a and b whose coefficients are elements of F
 
  • #5


Hurkyl said:
F[a] is the set of polynomials in b whose coefficients are elements of F[a]


Oh that makes sense-- it's sort of recursive... OK. God my book is so muddled and just starts talking about F[a] out of the blue like it's clear what that would mean.

Thank you so much no I can keep studying!

:rolleyes:
 

1. What is the difference between F[a][b] and F[a,b]?

The main difference between F[a][b] and F[a,b] is the way they are represented in mathematical notation. F[a][b] indicates a function that takes two inputs, a and b, while F[a,b] indicates a function of two variables, a and b. In other words, F[a][b] is a function that takes two values as parameters, while F[a,b] is a function that has two independent variables.

2. Can both notations be used interchangeably?

No, F[a][b] and F[a,b] cannot be used interchangeably. They have different meanings and represent different types of functions. Mixing up the notations can result in incorrect mathematical expressions and lead to errors in calculations.

3. When should I use F[a][b] and when should I use F[a,b]?

The choice between using F[a][b] and F[a,b] depends on the context and the specific function being represented. If the function takes two values as inputs, such as F(x,y), then F[a,b] would be the appropriate notation. If the function has two variables, such as f(x,y), then F[a][b] would be more suitable.

4. Is there a difference in how these notations are used in different fields of science?

Yes, there can be variations in how F[a][b] and F[a,b] are used in different fields of science. In mathematics, F[a][b] is commonly used to represent a function, while in physics and engineering, F[a,b] is often used to denote a physical quantity with two independent variables. It is important to understand the conventions in your specific field of study.

5. Are there any other notations for representing functions with two inputs or variables?

Yes, there are other notations that can be used to represent functions with two inputs or variables, such as F(a,b) or F(ab). These notations may vary depending on the context and the specific function being represented, but they all indicate a function of two variables or two inputs.

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