How to differentiate a function as injective and bijective? explain it by proper example.

A function is f:AB is said to be injective or one-one if different elements of A have different images in B. This means that for a₁, a₂ ∈ A and f(a₁), f(a₂) ∈ B,

f(a₁) = f(a₂) then a₁= a₂. or, a₁ a₂ f(a₁) f(a₂).

A function f:AB is said to be onto or surjective if each element in B is the f-image of atleast one element in A.

f:AB is onto iff for every b ∈ B, ∃ some a ∈ A such that f(a) = b.

An easy way to determine whether a function is onto or not is to check whether the range = codomain. if yes, then the function is onto else not.

A function which is one-one as well as onto is said to be bijective.

Eg: f(x) = x, if x is rational

0, if x is irrational

g(x) =0, if x is rational