Yes, so long as you keep in mind how σ is defined, as regards what equivalence class maps to what other equivalence class.Danielm said:Homework Statement
Let σ : Z_11 → Z_11 be given by σ([a]) = [5a + 3]). Prove that σ is bijective.
Homework Equations
The Attempt at a Solution
I am just wondering if I can treat σ as a normal function and prove that is bijective by using the definitions of one to one and onto.
Bijectivity in a function means that for every element in the domain, there is a unique element in the codomain, and vice versa. In other words, there is a one-to-one correspondence between the elements of the domain and the elements of the codomain.
Proving the bijectivity of a function is important because it ensures that the function is well-defined and has a unique inverse. It also guarantees that the function has a one-to-one correspondence between its domain and codomain, which is useful in various mathematical applications.
To prove the bijectivity of a function, you must show that the function is both injective (one-to-one) and surjective (onto). This can be done by showing that every element in the codomain has a unique preimage in the domain, and that every element in the codomain is mapped to by at least one element in the domain.
In the given example, we can prove the bijectivity of the function σ : Z_11 → Z_11 by showing that it is both injective and surjective. This can be done by substituting values from the domain and verifying that each element in the codomain has a unique preimage, and that every element in the codomain is mapped to by at least one element in the domain.
Providing a homework solution for proving the bijectivity of a function allows students to understand the concept better and apply it to similar problems. It also serves as a reference for future use and helps in the learning process.