The function σ : Z_11 → Z_11 defined by σ([a]) = [5a + 3] is proven to be bijective. The proof involves demonstrating that σ is both one-to-one and onto, adhering to the definitions of bijectivity. The key to the proof lies in understanding the mapping of equivalence classes within the modular arithmetic framework of Z_11. By establishing that each output corresponds uniquely to an input and that all possible outputs are covered, the bijectivity of σ is confirmed.
PREREQUISITESStudents of abstract algebra, mathematicians interested in modular arithmetic, and anyone studying the properties of functions in mathematical proofs.
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.