15. (a) (i) Let \( \quad(L, \leq) \quad \) be a lattice. For any \( a, b \in L \), \( a \leq b \Leftrightarrow a \wedge b=a \Leftrightarrow a \vee b=b \).

c) \( \begin{array}{l}\mathrm{U}=\{1,2,3, \ldots 10\} \text { is a given universal set. } O=\{1,3,5,7,9\} \text { and } \\ \mathrm{P}=\{\mathbf{3}, \mathbf{5}, 7\} \text { are any two given sets. } \\ \text { (i) Is the set } O \text { a subset of } U \text { ? Give reason. } \\ \text { (ii) Is the set } P \text { a proper or improper subset of } O \text { ? Give reason. } \\ \text { (iii) Show the sets } O \text { and } P \text { in a diagram. }\end{array} \).

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)

Q. 15 Find the eigenvalues and eigenvectors of the matrix \( \left[\begin{array}{ll}6 & 5 \\ 3 & 4\end{array}\right] \).

1. Representar en un diagrama de Venn el conjunto \( [A-(A \cap B)]^{C} \). Considere a) \( A \) subconjunto de \( B \) b) \( B \) subconjunto de \( A \)

Explain the difference between exponential growth and exponential decay.