Injective (One-to-One)
A function f: A โ B is injective if no two elements of A map to the same element of B.
Definition: f(aโ) = f(aโ) โ aโ = aโ
Example: f(x) = 2x is injective (different inputs give different outputs)
Counter: f(x) = xยฒ is NOT injective (f(2) = f(-2) = 4)Surjective (Onto)
A function f: A โ B is surjective if every element of B is the image of some element of A.
Definition: For every b โ B, there exists a โ A such that f(a) = b
Range = Codomain
Example: f: โ โ โ, f(x) = xยณ is surjective
Counter: f: โ โ โ, f(x) = eหฃ is NOT surjective (never negative)Bijective (One-to-One Correspondence)
A function is bijective if it is both injective and surjective. Bijective functions can be reversed.
f is bijective โบ f is injective AND f is surjective
Example: f: โ โ โ, f(x) = 2x + 1 is bijective
Injective: 2a+1 = 2b+1 โ a = b โ
Surjective: For any y, x = (y-1)/2 gives f(x) = y โ