Inverses and Injections
Inverse of a Relation
The inverse of a binary relation R is formed by swapping each ordered pair.
R⁻¹ = { (y, x) | (x, y) ∈ R }
Example: Inverse Function
Using earlier example f:
f⁻¹ = {(2,0),(3,1),(4,2),(5,3),(6,4)}
Injection (One-to-One)
A relation R is injective if both R and R⁻¹ are functions.
This means no two inputs give the same output.
Example: Injective Function
Function f is injective.
f = {(0,2),(1,3),(2,4),(3,5),(4,6)}
No repeated outputs → each input maps uniquely
Example: Not Injective
Function g is not injective.
g(0) = 3 and g(5) = 3
Different inputs give same output
Key Idea
Injective means: different inputs always give different outputs.