If $\mathrm{A} \times \mathrm{B}=\{(p, q),(p, r),(m, q),(m, r)\}$, find $\mathrm{A}$ and $\mathrm{B}$.

Step 1: Understand Cartesian Product

The Cartesian product of two sets, A and B, denoted as $A \times B$, 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 $A \times B$ must belong to set A. We examine the given set $A \times B = \{(p, q),(p, r),(m, q),(m, r)\}$ 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 $A \times B$ must belong to set B. We examine the given set $A \times B = \{(p, q),(p, r),(m, q),(m, r)\}$ 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 $\{q, r\}$.

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