Construct a multiplication table for modulo 9 on the set 2, 4, 6, 8. Then find the truth set of n n = n − 6 and evaluate (6 4) (4 2).

Here are the solutions to the problems.

2. (a) Construct a multiplication $\otimes$ table for modulo 9 on the set $\{2, 4, 6, 8\}$.

Step 1: Create a table and calculate each product $a \times b \pmod{9}$ for $a, b \in \{2, 4, 6, 8\}$.
For example, $2 \otimes 2 = (2 \times 2) \pmod{9} = 4 \pmod{9} = 4$.
$2 \otimes 6 = (2 \times 6) \pmod{9} = 12 \pmod{9} = 3$.
$6 \otimes 8 = (6 \times 8) \pmod{9} = 48 \pmod{9} = 3$.
$8 \otimes 8 = (8 \times 8) \pmod{9} = 64 \pmod{9} = 1$.

The completed multiplication table is:

$$\begin{array}{|c|c|c|c|c|} \hline \otimes & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 8 & 3 & 7 \\ \hline 4 & 8 & 7 & 6 & 5 \\ \hline 6 & 3 & 6 & 0 & 3 \\ \hline 8 & 7 & 5 & 3 & 1 \\ \hline \end{array}$$

2. (b) i. Using the table, find the truth set of $n \otimes n = n - 6$.

Step 1: Test each value of $n$ from the set $\{2, 4, 6, 8\}$ in the equation $n \otimes n = n - 6$.
For $n=2$: $2 \otimes 2 = 4$. $2 - 6 = -4$. $4 \neq -4$.
For $n=4$: $4 \otimes 4 = 7$. $4 - 6 = -2$. $7 \neq -2$.
For $n=6$: $6 \otimes 6 = 0$. $6 - 6 = 0$. $0 = 0$. This is true.
For $n=8$: $8 \otimes 8 = 1$. $8 - 6 = 2$. $1 \neq 2$.

Step 2: Identify the values of $n$ for which the equation holds true.
The only value for which the equation is true is $n=6$.
The truth set is $\boxed{\mathbf{\{6\}}}$.

2. (b) ii. Evaluate $(6 \otimes 4) \otimes (4 \otimes 2)$.

Step 1: Evaluate the expression inside the first parenthesis, $6 \otimes 4$, using the table.

$$6 \otimes 4 = 6$$

Step 2: Evaluate the expression inside the second parenthesis, $4 \otimes 2$, using the table.

$$4 \otimes 2 = 8$$

Step 3: Substitute these results back into the original expression and evaluate.

$$(6 \otimes 4) \otimes (4 \otimes 2) = 6 \otimes 8$$

From the table, $6 \otimes 8 = 3$.
The evaluation is $\boxed{\mathbf{3}}$.

3. (a) The scores obtained by 10 students in a test have the following statistics: $\sum x = 290$ and $\sum x^2 = 8469$. Calculate: