If A×B={(p,q),(p,r),(m,q),(m,r)}, find A and B.
Get the complete, step-by-step math solution for: "If {A} × {B}=\{(p, q),(p, r),(m, q),(m, r)\}, find {A} and {B}.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Understand Cartesian Product
The Cartesian product of two sets, A and B, denoted as A×B, is the set of all possible ordered pairs where the first element of each pair comes from set A and the second element comes from set B. Each element from A is paired with every element from B.
Step 2: Identify Elements of Set A
From the definition of the Cartesian product, the first element of every ordered pair in A×B must belong to set A. We examine the given set A×B={(p,q),(p,r),(m,q),(m,r)} and collect all unique first elements.
Step 3: Determine Set A
The first elements in the given ordered pairs are p, p, m, m. Collecting the unique first elements gives us set A as {p,m}.
Step 4: Identify Elements of Set B
Similarly, the second element of every ordered pair in A×B must belong to set B. We examine the given set A×B={(p,q),(p,r),(m,q),(m,r)} and collect all unique second elements.
Step 5: Determine Set B
The second elements in the given ordered pairs are q, r, q, r. Collecting the unique second elements gives us set B as {q,r}.