Questions: Functions as Sets and Their Characteristics

Question 1

Let A = {1, 2} and B = {a, b, c}. Which of the following subsets of A × B is NOT a valid function from A to B?

• A. {(1, a), (2, b)}
• B. {(1, a), (1, b), (2, c)}
• C. {(1, c), (2, c)}
• D. {(1, a), (2, a)}

Answer: (B)

A function requires each element of the domain to map to exactly one element of the codomain. In option B, element 1 is paired with both a and b — violating the 'exactly one' requirement. Options A, C, and D each assign every element of A to a single element of B, so all three are valid functions (even though C and D are not injective).

Question 2

Two functions f and g from A to B are the same function if and only if they have the same domain A and the same codomain B.

• T. True
• F. False

Answer: False

Domain and codomain only specify the type signature — f: A → B and g: A → B. Two functions with the same domain and codomain can differ in which element of B each a ∈ A maps to. Functions are equal only if f(a) = g(a) for every a ∈ A, meaning their underlying subsets of A × B are identical.

Question 3

What is the difference between the codomain and the image of a function f: A → B?

Model answer: The codomain B is the declared target set — all allowable outputs. The image is {f(a) | a ∈ A}, the set of outputs actually achieved. The image is always a subset of the codomain, but they are equal only when f is surjective.

Conflating codomain and image is a common error. For example, f: ℝ → ℝ defined by f(x) = x² has codomain ℝ but image [0, ∞) — negative numbers are in the codomain but never output. The distinction matters for defining surjectivity and for inverses.