Sets & Cartesian Product — If P = {1, 2, 3} and Q = {4, 5}, determine the Cartesian product P × Q (ordered pairs with first element from P and second from Q).

Introduction / Context:
Cartesian product P × Q is the set of all ordered pairs (p, q) where p belongs to the first set P and q belongs to the second set Q. Order matters: (p, q) is different from (q, p) unless p = q and both lie in both sets. Here P has three elements and Q has two elements.

Given Data / Assumptions:

• P = {1, 2, 3}
• Q = {4, 5}
• Ordered pairs are formed with first element from P, second from Q.

Concept / Approach:
List systematically by fixing the first coordinate from P and pairing with each element of Q. For p = 1: (1, 4), (1, 5). For p = 2: (2, 4), (2, 5). For p = 3: (3, 4), (3, 5). Altogether there are |P| × |Q| = 3 × 2 = 6 ordered pairs.

Step-by-Step Solution:
Pairs with 1: (1, 4), (1, 5)Pairs with 2: (2, 4), (2, 5)Pairs with 3: (3, 4), (3, 5)Collect: { (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) }

Verification / Alternative check:
Count check: exactly 6 pairs = 3 × 2, no repeats and no pair starts with 4 or 5 (since those are not in P).

Why Other Options Are Wrong:
Option a includes (4, 5), which is invalid because 4 ∉ P for the first coordinate; it also omits (1, 4). Option b similarly contains (4, 5). Option d duplicates (3, 5) and omits (3, 4). Option e is just a reordering of the correct set (also valid content-wise), but option c is the canonical, clean listing without duplication or extraneous items, so it is the best answer here.

Common Pitfalls:
Mixing up the order of coordinates or assuming sets are unordered pairs; forgetting that every element of P must pair with every element of Q.

Final Answer:
{ (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) }