Questions: Cartesian Product

Question 1

Let A = {1, 2, 3} and B = {a, b}. What is |A × B|?

• A. 5, because |A| + |B| = 3 + 2 = 5
• B. 6, because |A| · |B| = 3 · 2 = 6
• C. 8, because A × B and B × A together have 12 pairs but we remove duplicates
• D. 9, because the product of the largest elements is 3 · 3 = 9

Answer: (B)

The size formula for Cartesian products is multiplicative: |A × B| = |A| · |B|. For each of the 3 elements in A, you form a pair with each of the 2 elements in B, giving 3 · 2 = 6 ordered pairs: {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}. The additive answer (5) confuses union with product — union combines elements, product forms pairs. This multiplicative structure extends to more sets: |A × B × C| = |A| · |B| · |C|.

Question 2

Which statement correctly distinguishes the set {a, b} from the ordered pair (a, b)?

• A. They are the same object — sets and ordered pairs represent identical mathematical structures
• B. In a set, {a, b} = {b, a}; in an ordered pair, (a, b) ≠ (b, a) unless a = b
• C. Sets can only contain numbers, while ordered pairs can contain any objects
• D. Ordered pairs are just sets with exactly two elements — the notation is different but the meaning is identical

Answer: (B)

The key distinction is that sets are unordered: {a, b} and {b, a} are the same set. Ordered pairs are ordered: (a, b) and (b, a) are different unless a = b. This distinction is what makes the Cartesian product useful. A × B and B × A contain the same element combinations but different ordered pairs: (a, b) ∈ A × B while (b, a) ∈ B × A. Order-sensitivity is essential for defining relations and functions, where which element 'maps to' which matters.

Question 3

A binary relation from set A to set B is formally defined as a subset of the Cartesian product A × B.

• T. True
• F. False

Answer: True

This is the formal set-theoretic definition of a relation. A × B is the set of all possible ordered pairs (a, b). Any subset R ⊆ A × B specifies exactly which pairs stand in the relation — (a, b) ∈ R means 'a is related to b.' For example, the 'less than' relation on {1, 2, 3} is the subset {(1,2), (1,3), (2,3)} ⊆ {1,2,3} × {1,2,3}. Functions are then defined as special relations: subsets of A × B where every element of A appears as a first coordinate exactly once.

Question 4

For any two sets A and B, A × B = B × A.

• T. True
• F. False

Answer: False

In general, A × B ≠ B × A. A × B consists of pairs (a, b) with a ∈ A and b ∈ B; B × A consists of pairs (b, a) with b ∈ B and a ∈ A. These are different ordered pairs unless a = b. For example, if A = {1} and B = {x}, then A × B = {(1,x)} while B × A = {(x,1)} — completely different sets. The only exceptions are when A = B (in which case A × A = A × A trivially) or when one of the sets is empty (both products are then empty).

Question 5

Why is the concept of an ordered pair — rather than just a set of two elements — necessary for defining functions and relations?

Model answer: Functions and relations track which element is the input and which is the output (or which is related to which). A set {a, b} loses this directionality: {a, b} = {b, a}, so we cannot tell whether 'a maps to b' or 'b maps to a.' An ordered pair (a, b) preserves the distinction: (a, b) ≠ (b, a) unless a = b. For example, the function f(2) = 5 must be recorded as (2, 5) — a different relationship than f(5) = 2, which is (5, 2). If we used unordered sets, we would lose the ability to represent which value was the input. Order is what makes the formal definition of function — as a set of input-output pairs — meaningful.

The Cartesian product, by generating ordered pairs, provides the formal scaffolding that makes every downstream concept in mathematics (functions, relations, graphs, matrices, coordinate geometry) rigorously definable. The concept is deceptively simple, but it is the foundation on which most of formal mathematics is built.