Ker(phi) = {0}, then phi injective?

Thread starter ctrlaltdelete
Start date Aug 18, 2008
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Injective Phi

In summary, the notation "Ker(phi) = {0}" refers to the kernel of a function phi, which is the set of all inputs that result in an output of 0. A function is injective if each element in the co-domain has a unique element in the domain that maps to it. Having a kernel of {0} in an injective function means that there are no non-zero elements in the domain that map to the identity element in the co-domain, making it easier to reverse the function. However, a function can still be injective even if its kernel is not equal to {0}. The notation "Ker(phi) = {0}" is directly related to the concept of one-to-one functions, indicating

Aug 18, 2008

ctrlaltdelete

Hi all,

Can anyone point to an explanation of why if ker(phi) = {0}, then phi is injective?

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Aug 18, 2008

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Homework Helper

Is "phi" a linear map?

Suppose that phi(x) = phi(y), and use some properties of linear maps to show that x = y.

1. What does "Ker(phi) = {0}" mean?

The notation "Ker(phi) = {0}" refers to the kernel of the function phi. The kernel is a set of all elements in the domain of phi that map to the identity element in the co-domain. In simpler terms, it is the set of all inputs that result in an output of 0 when plugged into phi.

2. How is the function phi injective?

A function is injective if each element in the co-domain has a unique element in the domain that maps to it. In this case, since the kernel is equal to {0}, it means that there is only one element in the domain that maps to the identity element in the co-domain. Therefore, the function phi is injective.

3. What is the significance of having a kernel of {0} in an injective function?

The kernel being equal to {0} in an injective function means that there are no non-zero elements in the domain that map to the identity element in the co-domain. This shows that the function is one-to-one and there are no duplicates in the output, making it easier to reverse the function.

4. Can a function be injective if its kernel is not equal to {0}?

Yes, a function can still be injective even if its kernel is not equal to {0}. As long as each element in the co-domain has a unique element in the domain that maps to it, the function is considered injective. However, having a kernel of {0} makes it easier to prove that the function is injective.

5. How is "Ker(phi) = {0}" related to the concept of one-to-one functions?

The notation "Ker(phi) = {0}" is directly related to the one-to-one concept in functions. It means that there is no non-zero element in the domain that maps to the same element in the co-domain, making the function one-to-one. This shows that the function has a unique inverse and can be reversed easily.