1. **Problem Statement:** Draw addition (\oplus) and multiplication (\otimes) tables for the set $\{2,4,6,8\}$ modulo 9. Then, from these tables: (i) Evaluate $(4 \oplus 6) \oplus (2 \otimes 4)$. (ii) Find $n$ such that $3n \otimes 8 = 3$. (iii) Find the truth set where $(n \otimes n)$ is odd. 2. **Formulas and Rules:** - Addition modulo 9: $a \oplus b = (a + b) \bmod 9$ - Multiplication modulo 9: $a \otimes b = (a \times b) \bmod 9$ - Odd numbers modulo 9 are those congruent to 1, 3, 5, or 7. 3. **Construct Addition Table $\oplus$ modulo 9:** Calculate $(a + b) \bmod 9$ for $a,b \in \{2,4,6,8\}$: - $2 \oplus 2 = 4$ - $2 \oplus 4 = 6$ - $2 \oplus 6 = 8$ - $2 \oplus 8 = 10 \bmod 9 = 1$ - $4 \oplus 2 = 6$ - $4 \oplus 4 = 8$ - $4 \oplus 6 = 10 \bmod 9 = 1$ - $4 \oplus 8 = 12 \bmod 9 = 3$ - $6 \oplus 2 = 8$ - $6 \oplus 4 = 10 \bmod 9 = 1$ - $6 \oplus 6 = 12 \bmod 9 = 3$ - $6 \oplus 8 = 14 \bmod 9 = 5$ - $8 \oplus 2 = 10 \bmod 9 = 1$ - $8 \oplus 4 = 12 \bmod 9 = 3$ - $8 \oplus 6 = 14 \bmod 9 = 5$ - $8 \oplus 8 = 16 \bmod 9 = 7$ Addition table $\oplus$: \begin{array}{c|cccc} \oplus & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 6 & 8 & 1 \\ 4 & 6 & 8 & 1 & 3 \\ 6 & 8 & 1 & 3 & 5 \\ 8 & 1 & 3 & 5 & 7 \end{array} 4. **Construct Multiplication Table $\otimes$ modulo 9:** Calculate $(a \times b) \bmod 9$ for $a,b \in \{2,4,6,8\}$: - $2 \otimes 2 = 4$ - $2 \otimes 4 = 8$ - $2 \otimes 6 = 12 \bmod 9 = 3$ - $2 \otimes 8 = 16 \bmod 9 = 7$ - $4 \otimes 2 = 8$ - $4 \otimes 4 = 16 \bmod 9 = 7$ - $4 \otimes 6 = 24 \bmod 9 = 6$ - $4 \otimes 8 = 32 \bmod 9 = 5$ - $6 \otimes 2 = 12 \bmod 9 = 3$ - $6 \otimes 4 = 24 \bmod 9 = 6$ - $6 \otimes 6 = 36 \bmod 9 = 0$ - $6 \otimes 8 = 48 \bmod 9 = 3$ - $8 \otimes 2 = 16 \bmod 9 = 7$ - $8 \otimes 4 = 32 \bmod 9 = 5$ - $8 \otimes 6 = 48 \bmod 9 = 3$ - $8 \otimes 8 = 64 \bmod 9 = 1$ Multiplication table $\otimes$: \begin{array}{c|cccc} \otimes & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 8 & 3 & 7 \\ 4 & 8 & 7 & 6 & 5 \\ 6 & 3 & 6 & 0 & 3 \\ 8 & 7 & 5 & 3 & 1 \end{array} 5. **Evaluate (i) $(4 \oplus 6) \oplus (2 \otimes 4)$:** - From addition table: $4 \oplus 6 = 1$ - From multiplication table: $2 \otimes 4 = 8$ - Now $1 \oplus 8 = (1 + 8) \bmod 9 = 9 \bmod 9 = 0$ 6. **Solve (ii) Find $n$ such that $3n \otimes 8 = 3$:** - Note $3n$ means $3 \otimes n$ modulo 9. - Compute $3 \otimes n$ for $n \in \{2,4,6,8\}$: - $3 \otimes 2 = 6$ - $3 \otimes 4 = 3$ - $3 \otimes 6 = 0$ - $3 \otimes 8 = 6$ - Now multiply each by $8$ modulo 9: - For $n=2$: $(3 \otimes 2) \otimes 8 = 6 \otimes 8 = (6 \times 8) \bmod 9 = 48 \bmod 9 = 3$ - For $n=4$: $3 \otimes 8 = 3 \otimes 8 = 6$ (not 3) - For $n=6$: $0 \otimes 8 = 0$ - For $n=8$: $6 \otimes 8 = 3$ - So $n=2$ or $n=8$ satisfy $3n \otimes 8 = 3$. 7. **Find (iii) truth set where $(n \otimes n)$ is odd:** - Calculate $n \otimes n$ for each $n$: - $2 \otimes 2 = 4$ (even) - $4 \otimes 4 = 7$ (odd) - $6 \otimes 6 = 0$ (even) - $8 \otimes 8 = 1$ (odd) - Odd results are for $n=4$ and $n=8$. **Final answers:** (i) $(4 \oplus 6) \oplus (2 \otimes 4) = 0$ (ii) $n = 2$ or $n = 8$ (iii) Truth set $= \{4, 8\}$