This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Understand the given information. We are given that the number of common elements between A × B and B × A is 25. We need to find the number of elements common to A and B, i.e., n(A B). Step 2: Determine the structure of common elements. An ordered pair (x, y) is common to A × B and B × A if and only if: • (x, y) A × B x A and y B • (x, y) B × A x B and y A For (x, y) to be in both sets, we must have x A and x B, which means x A B. Similarly, y B and y A, which means y A B. Therefore, the set of common elements is (A B) × (A B). Step 3: Calculate the number of common elements. Let n(A B) = k. The number of elements in (A B) × (A B) is n((A B) × (A B)) = n(A B) · n(A B) = k · k = k^2. We are given that this number is 25. k^2 = 25 Step 4: Solve for k. Taking the square root of both sides: k = sqrt(25) Since k represents the number of elements, it must be a non-negative integer. k = 5 Thus, the number of elements common to A and B is 5. Step 5: Select the correct option. The calculated value is 5, which corresponds to option (C). The final answer is C Send me the next one 📸