Let \(A = \{1, 2, 3, \ldots, 10\}\) and let R be the equivalence relation on \(A \times A\) defined by \((a, b)\,R\,(c, d)\) if \(ad = bc\). The number of elements in the equivalence class \([(1, 2)]\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 5
A pair (c, d) of \(A \times A\) is in \([(1, 2)]\) when \((1, 2)\,R\,(c, d)\), that is,
\(1 \times d = 2 \times c\)
\(d = 2c\)
Both c and d must lie in A, so \(2c \le 10\), which gives \(c = 1, 2, 3, 4, 5\):
\([(1, 2)] = \{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\}\)
So the class has 5 elements.