Injective, Surjective, Bijective

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Functions can have special properties:

Injective (one-to-one): f : X → Y is injective if:

For all x₁, x₂ ∈ X, f(x₁) = f(x₂) ⇒ x₁ = x₂. Equivalently: different inputs give different outputs.
Surjective (onto): f : X → Y is surjective if:

For every y ∈ Y, there exists x ∈ X such that f(x) = y. Every element of the codomain is actually hit.
Bijective: f is bijective if it is both injective and surjective. This means f is a perfect matching between X and Y.

Proof patterns:

To prove injective: assume f(x₁) = f(x₂) and show x₁ = x₂.
To prove surjective: start with an arbitrary y in Y and solve for x in terms of y.

TL;DR — key idea

Injective means no two inputs share an output; surjective means every codomain element is hit; bijective means both.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
For the function f : ℤ → ℤ given by f(n) = 2n + 3: 1. Prove or disprove that f is injective. 2. Prove or disprove that f is surjective. Be explicit with quantifiers: "for all" and "there exists."

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.