Questions: Injective, Surjective, and Bijective Functions

Question 1

Consider f: ℝ → ℝ defined by f(x) = x². Is this function surjective?

• A. Yes — every positive real number has a square root, so the range includes almost everything
• B. No — negative numbers are never outputs of x², so f does not cover the full codomain ℝ
• C. Yes — f is defined for all real inputs, so it must hit all real outputs
• D. It depends — surjectivity requires checking injectivity first

Answer: (B)

Surjectivity requires that every element of the codomain be reached. The codomain is ℝ (all reals), but x² ≥ 0 for all real x — negative numbers are never outputs. So f: ℝ → ℝ with f(x) = x² is NOT surjective. Note the contrast: if we restrict the codomain to [0, ∞), declaring f: ℝ → [0, ∞), the same formula becomes surjective. Surjectivity is a relationship between the range and the declared codomain, not just a property of the formula.

Question 2

Which of the following correctly describes a bijection between two sets A and B?

• A. A function where every element of A maps to a unique element of B, and |A| = |B|
• B. A function that is both injective and surjective: no two inputs share an output, and every codomain element is reached
• C. Any function from A to B that has an algebraically computable inverse formula
• D. A function where every element of A is in the range of B

Answer: (B)

A bijection is precisely the combination of injectivity (no collisions — distinct inputs map to distinct outputs) and surjectivity (no gaps — every codomain element is reached). This creates a perfect one-to-one pairing. Option A is tempting but wrong: |A| = |B| is true for finite sets with a bijection, but for infinite sets |A| = |B| is defined *by* the existence of a bijection, making it circular. Option C is wrong: some bijections (e.g., Cantor's pairing function) have inverses that aren't simple formulas.

Question 3

The function n ↦ 2n from ℤ to ℤ (mapping each integer to its double) is injective but not surjective.

• T. True
• F. False

Answer: True

Injective: if 2m = 2n, then m = n — distinct inputs give distinct outputs. Not surjective: the odd integers (1, 3, 5, ...) are in the codomain ℤ but are never outputs of 2n. This illustrates that injectivity and surjectivity are independent properties: a function can have one without the other.

Question 4

A bijection between an infinite set and one of its proper subsets is impractical — it would violate the principle that a whole is greater than its parts.

• T. True
• F. False

Answer: False

This is false — and it is the defining counterintuitive feature of infinite sets. The function n ↦ 2n bijects ℤ with the even integers, a proper subset of ℤ. Dedekind used this property as the *definition* of an infinite set: a set is infinite if and only if it can be put into bijection with a proper subset of itself. The intuition that 'whole > part' holds for finite sets but breaks down for infinite ones.

Question 5

Why can a bijection exist between an infinite set and one of its proper subsets, and what does this reveal about infinite cardinality?

Model answer: For infinite sets, bijections can exist between a set and a proper subset because the 'size' of an infinite set is not exhausted by removing a finite (or even infinite) number of elements in the right way. For example, n ↦ 2n pairs every integer with a unique even integer — no element on either side is left unpaired — establishing a bijection despite the even integers being a proper subset of the integers. This reveals that infinite cardinality is not governed by the part-whole principle. Two infinite sets have the same cardinality if and only if a bijection exists between them, and this definition allows proper subsets to be 'equally large.'

This distinction — finite intuitions failing for infinite sets — is the central insight of Cantorian set theory. It also explains why bijections are the right tool for comparing sizes: for finite sets, a bijection between A and B iff |A| = |B| matches our counting intuition; for infinite sets, bijection is the only coherent notion of 'same size.' Understanding this is essential for studying countability and uncountability.