[The most general such \( R \) is \( U \sqrt{A} \), where \( U \) is some isometry.] 6- Let \( x_{1}, \ldots, x_{n} \) be \( n \) vectors. Let \( G \) be a matrix with elements \( G_{i j}=\left(x_{i}, x_{j}\right) \). Show that \( G \) is a positive matrix. [ \( G \) is called the Gram matrix.]

Why is understanding your available resources essential for effectively assisting customers? it allows you to avoid dealing with difficult customers that you can always find a solution without needing further assistance it prepares you for unexpected inquiries and helps you anticipate needs to delegate tasks to other team members

Part-III: Choose the best answer from given choice (Ipt eacil 11. Let \( B=\{\emptyset,\{\emptyset,\{\emptyset\}\}\} \). Then the number of proper subset is \( \begin{array}{llll}\text { A. } 8 & \text { B. } 7 & \text { C. } 3 & \text { D. } 4\end{array} \)

68. Which step in the market research process allows conclusions to be drawn an recommendations to be formulated? A. Data collection B. Data analysis C. Qualitative analysis D. Quantitative analysis

Part-I: Write true if the statement is true or false if the statement is false. (1pt each) 1. If \( A=\{0,1,2\} \) and \( B=\{x \in R \mid 0 \leq x \leq 2\} \), then \( A=B \). 2. \( \{\emptyset\} \) is both the element and the subset of \( \{\emptyset,\{\emptyset\}\} \). 3. If the truth value of \( p \Rightarrow q \) is \( F \), then the truth values of \( (\neg p \wedge q) \Leftrightarrow(p \vee q) \) is false. 4. If \( A=\{x \in Z \mid x \) is divisible by 2\( \} \) and \( B=\{y \in Z \mid y \) is divisible by 3\( \} \), then \( A \cap B \) is \( \emptyset \).

Part-III: Choose the best answer from given choice (1pt eacil) 11. Let \( \mathrm{B}=\{\emptyset,\{\emptyset,\{\emptyset\}\}\} \). Then the number of proper subset is \( \begin{array}{llll}\text { A. } 8 & \text { B. } 7 & \text { C. } 3 & \text { D. } 4\end{array} \)