CBecause 3 and 5 are different numbers, so any pairing must respect their natural ordering
DBecause sets preserve the order in which elements were inserted
Sets are inherently unordered — {3,5} and {5,3} are identical as sets. To encode order within set theory, Kuratowski defined (a,b) = {{a},{a,b}}. The first component a appears as a singleton {a}, unique to the pair encoding, which distinguishes it from b. You can verify: (3,5) = {{3},{3,5}} while (5,3) = {{5},{5,3}}, and these are different sets because their singletons differ. The equality rule (a,b) = (c,d) iff a=c and b=d follows directly.
Question 2 Multiple Choice
If |A| = 4 and |B| = 3, how many elements does A × B contain?
A7 — the sum of the sizes
B12 — the product of the sizes
CIt depends on how many elements A and B share
D24 — the number of ways to arrange all elements
|A × B| = |A| · |B| = 4 · 3 = 12. For each of the 4 elements of A, you form an ordered pair with each of the 3 elements of B independently — giving 4 × 3 = 12 distinct pairs. Shared elements between A and B are irrelevant because ordered pairs track both components: even if a ∈ A ∩ B, the pair (a,a) is a valid element of A × B, distinct from (a,x) for any x ≠ a. The product formula follows from the independent choice of first and second component.
Question 3 True / False
For any two sets A and B, A × B = B × A.
TTrue
FFalse
Answer: False
A × B contains pairs (a,b) with a ∈ A first and b ∈ B second. B × A contains pairs (b,a) with b ∈ B first and a ∈ A second. Unless A = B (or one is empty), these are different sets because (a,b) ≠ (b,a) when a ≠ b. For example, if A = {1} and B = {x}, then A × B = {(1,x)} while B × A = {(x,1)}, and (1,x) ≠ (x,1). The order of the sets in the product determines which element comes first in each pair.
Question 4 True / False
A function f: A → B can be formally defined as a subset of the Cartesian product A × B.
TTrue
FFalse
Answer: True
A function f: A → B is formally a relation (a subset R ⊆ A × B) with the additional constraint that every element of A appears as a first component exactly once: for each a ∈ A, there is exactly one pair (a,b) ∈ R. This 'exactly one' condition captures the requirement that a function assigns a unique output to every input. The Cartesian product provides the universe of all possible input-output pairs; the function selects the specific subset satisfying the uniqueness condition.
Question 5 Short Answer
Why is it necessary to formally encode ordered pairs as sets (using the Kuratowski definition or similar), rather than simply treating them as a new primitive notion alongside sets?
Think about your answer, then reveal below.
Model answer: Set theory aims to provide a single unified foundation in which all mathematical objects are sets. If ordered pairs were treated as a separate primitive, the foundation would require additional axioms and a new sort of entity. By encoding (a,b) as {{a},{a,b}}, ordered pairs become sets constructible from existing axioms, keeping the foundational theory minimal. The encoding works as long as it satisfies the key property: (a,b) = (c,d) if and only if a = c and b = d.
The choice of encoding is not unique — other encodings exist — but Kuratowski's is the standard. What matters is that some encoding exists, ensuring ordered pairs don't require a new axiom. Once ordered pairs are sets, Cartesian products are sets, and then relations and functions are sets, allowing all of mathematics to be developed within the single framework of ZFC set theory.