Q5. Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2) and (z, 1) are in A × B, find A and B, where x, y and z are distinct elements (NCERT Exercise 2.1, Q9).
The first entries x, y, z are three distinct elements of A, and A has only 3, so A = {x, y, z}. The second entries 1 and 2 are in B, which has 2 elements, so B = {1, 2}.
Q6. A = {1, 2, 3} and B = {3, 4}. List (A × B) ∩ (B × A) and check its size against n(A ∩ B)2.
The common pairs need both entries in A ∩ B = {3}, so (A × B) ∩ (B × A) = {(3, 3)}: 1 pair, and n(A ∩ B)2 = 12 = 1.
Q7. n(A) = 4 and n(B) = 5, and A and B have exactly 2 elements in common. How many elements do A × B and B × A have in common?
(A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B), which has 2 × 2 = 4 elements.
Q8. If A = {1, 2} and B = {a, b, c}, how many elements does A × B × A have? How many of them have the same first and last entry?
n(A × B × A) = 2 × 3 × 2 = 12. For the first and last entries to match, choose the first entry (2 ways) and the middle one (3 ways): 6 triplets.
Q9. Let A = {1, 2, 3}. How many points of A × A lie on the line y = x, and how many lie above it (y > x)?
A × A has 9 points. On the line: (1, 1), (2, 2), (3, 3), so 3. The other 6 split equally between the two sides, because swapping the entries reflects a point in y = x: 3 lie above it, namely (1, 2), (1, 3) and (2, 3).