Questions: Injective, Surjective, and Bijective Functions

Question 1

Consider f: ℝ → ℝ defined by f(x) = x². A student claims this function is surjective because 'every input produces an output.' What is wrong with this reasoning?

• A. The student is correct — every real input does produce a real output, so the function is surjective
• B. Surjectivity requires every element of the codomain to be hit by some input — but negative reals are never outputs of f(x) = x², so f is not surjective from ℝ to ℝ
• C. The student's reasoning works for ℝ → ℝ but would fail for ℝ → [0, ∞)
• D. The function is not surjective because it is also not injective

Answer: (B)

Surjectivity is a claim about the codomain, not about the domain. The codomain is ℝ (all reals), but f(x) = x² only produces non-negative outputs — no negative number is ever hit. A student who confuses 'every input has an output' (true of every function, by definition) with 'every output is hit by some input' (surjectivity) is conflating the definition of a function with the additional property of surjectivity. Note that f: ℝ → [0, ∞) defined by the same rule *is* surjective — the codomain is part of the function's specification.

Question 2

Let f: ℕ → ℕ be defined by f(n) = 2n. Which properties does this function have?

• A. Injective only — distinct inputs produce distinct outputs, but odd natural numbers are never hit
• B. Surjective only — every element of ℕ is an even number times two
• C. Bijective — every natural number is the image of exactly one natural number
• D. Neither injective nor surjective

Answer: (A)

f(n) = 2n is injective: if 2n₁ = 2n₂ then n₁ = n₂, so no two distinct inputs share an output. But it is not surjective onto ℕ: the number 3 (or any odd number) is never an output of f. This is a classic illustration that injectivity and surjectivity are independent properties — a function can have one without the other. It also shows that the codomain matters: f: ℕ → {even natural numbers} defined by the same rule would be bijective.

Question 3

A bijection f: A → B implies that the inverse function f⁻¹: B → A exists as a well-defined function.

• T. True
• F. False

Answer: True

A function has a well-defined inverse exactly when it is bijective. Injectivity ensures that each output came from exactly one input (so the inverse rule 'go back to where you came from' is unambiguous). Surjectivity ensures that every element of B has a preimage at all (so the inverse is defined on all of B). A function that is injective but not surjective has a partial inverse; one that is surjective but not injective fails to have an inverse because some outputs came from multiple inputs.

Question 4

Whether a function is surjective depends primarily on its rule of assignment and is unaffected by how the codomain is defined.

• T. True
• F. False

Answer: False

Surjectivity is explicitly codomain-dependent. The same rule f(x) = x² defines a non-surjective function from ℝ to ℝ (negative reals are never hit) but a surjective function from ℝ to [0, ∞) (every non-negative real is hit). Changing only the codomain, without changing the rule, can turn a non-surjective function into a surjective one. This is why the codomain is treated as part of the function's definition, not just a background assumption.

Question 5

Explain why bijections are the correct tool for comparing the sizes of infinite sets, and give an example illustrating the key idea.

Model answer: A bijection establishes a perfect one-to-one correspondence between two sets: every element of each set is matched with exactly one element of the other, with no leftovers on either side. Two sets have the same cardinality — the same 'size' in the most fundamental sense — if and only if a bijection exists between them. For finite sets this matches ordinary counting. For infinite sets it produces surprising results: the function f(n) = 2n is a bijection between ℕ and the even natural numbers, showing these sets have the same cardinality even though one is a proper subset of the other. This is what it means for infinite sets to have the same size.

The example of ℕ and the even numbers is the classic illustration of Cantor's insight that 'same size' for infinite sets cannot mean 'same number of elements' in the ordinary sense — it must mean 'there is a bijection.' This is why bijections, and not counting, are the fundamental tool in cardinality theory.