Proving Ideal Property of f(x)=0 for Every Rational x in $\mathcal{F}(\mathcal{R})$

In summary, the question is asking to prove that two sets, one consisting of functions that have a constant value of 0 for all rational numbers, and the other consisting of functions that have a constant value of 0 at x=0, are both ideals of the ring $\mathcal{F}(\mathcal{R})$. There is a question about what the multiplicative operation is within the ring, with a suggestion that it may be composition of functions rather than standard multiplication on real numbers. However, it is clarified that the standard multiplication on real numbers is indeed the correct operation. The task is to prove that the two sets are ideals using pointwise multiplication.
  • #1
Kiwi1
108
0
I am asked:

Prove that each of the following is an ideal of $\mathcal{F}(\mathcal{R})$:
a. The set of all f such that f(x)=0 for every rational x
b. The set of all f such that f(0)=0

My question is how do I know what the multiplicative operation is within the ring? Is multiplication the standard multiplication on real numbers or is it composition of functions?

I would expect it to be composition of functions but then if I choose g(x)=1 then I get g(f(x))=g(0) for any real x and this is not generally zero. So it must be the multiplication on reals.

Am I just supposed to know that?
 
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  • #2
Hi Kiwi,

I don't know where this exercises come from, but it must be specified what the product is.

It is usual to consider $\mathcal{F}(\mathbb{R})$ the ring of real valued functions with pointwise product, and with this product your statements are true (they aren't with composition).

Try to prove it with pointwise multiplication and let us know if you encounter any other problem. ;)
 

FAQ: Proving Ideal Property of f(x)=0 for Every Rational x in $\mathcal{F}(\mathcal{R})$

What is the definition of f(x)=0 for every rational x in F(R)?

The statement "f(x)=0 for every rational x in F(R)" means that for any rational number x in the set of all real numbers, the output or value of the function f(x) is always equal to zero.

Why is it important to prove the ideal property of f(x)=0?

Proving the ideal property of f(x)=0 is important because it helps establish the behavior and characteristics of the function. It also allows us to accurately predict and analyze the behavior of the function for a given input.

How can we prove the ideal property of f(x)=0?

To prove the ideal property of f(x)=0, we can use mathematical techniques such as proof by contradiction, mathematical induction, or direct proof. These methods involve logically and rigorously showing that the function indeed satisfies the ideal property for every rational input.

What is the significance of proving the ideal property of f(x)=0 for every rational x?

Proving the ideal property of f(x)=0 for every rational x is significant because it helps us understand the relationship between rational numbers and the function. It also allows us to generalize the behavior of the function to other types of numbers, such as irrational numbers.

Can we prove the ideal property of f(x)=0 for all types of functions?

Yes, the ideal property of f(x)=0 can be proved for all types of functions, as long as the function is defined for all rational numbers. However, the method of proof may vary depending on the specific characteristics of the function.

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