Given set - \(\{1, 2, 3, 4\}\).
To form the smallest equivalence relation on this set that includes \((1, 2)\) and \((1, 3)\), we need to ensure that \(R\) is reflexive, symmetric, and transitive.
Step 1. Reflexive pairs: \((1, 1), (2, 2), (3, 3), (4, 4)\)
Step 2. Pairs to satisfy given conditions and transitivity:
- Since \((1, 2) \in R\) and \((1, 3) \in R\), we need \((2, 3) \in R\) for transitivity.
- For symmetry, include \((2, 1), (3, 1), (3, 2)\).
Step 3. Final set of pairs: \(R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}\).
Thus, the number of elements in \(R\) is 10.
The Correct Answer is: 10
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: