1. **Problem Statement:** Draw addition ⊕ and multiplication ⊗ tables for the set {2, 4, 6, 8} modulo 9. Evaluate (4 ⊕ 6) ⊕ (2 ⊗ 4), find n such that 3n ⊗ 8 = 3, and find the truth set where (n ⊗ n) is odd. 2. **Addition and Multiplication Modulo 9:** Addition modulo 9 means adding two numbers and then taking the remainder when divided by 9. Multiplication modulo 9 means multiplying two numbers and then taking the remainder when divided by 9. 3. **Addition Table (⊕):** Calculate each sum modulo 9: - 2⊕2=4, 2⊕4=6, 2⊕6=8, 2⊕8=1 - 4⊕2=6, 4⊕4=8, 4⊕6=1, 4⊕8=3 - 6⊕2=8, 6⊕4=1, 6⊕6=3, 6⊕8=5 - 8⊕2=1, 8⊕4=3, 8⊕6=5, 8⊕8=7 4. **Multiplication Table (⊗):** Calculate each product modulo 9: - 2⊗2=4, 2⊗4=8, 2⊗6=3, 2⊗8=7 - 4⊗2=8, 4⊗4=7, 4⊗6=6, 4⊗8=5 - 6⊗2=3, 6⊗4=6, 6⊗6=0, 6⊗8=3 - 8⊗2=7, 8⊗4=5, 8⊗6=3, 8⊗8=1 5. **Evaluate (4 ⊕ 6) ⊕ (2 ⊗ 4):** - 4 ⊕ 6 = 1 (from addition table) - 2 ⊗ 4 = 8 (from multiplication table) - Then (4 ⊕ 6) ⊕ (2 ⊗ 4) = 1 ⊕ 8 = 0 modulo 9 6. **Find n such that 3n ⊗ 8 = 3:** - 3n means 3 multiplied by n modulo 9. - We want (3n) ⊗ 8 ≡ 3 (mod 9). - Test n in {2,4,6,8}: - For n=2: 3*2=6; 6⊗8=3 (from multiplication table), so n=2. 7. **Find truth set where (n ⊗ n) is odd:** - Calculate n⊗n for each n: - 2⊗2=4 (even) - 4⊗4=7 (odd) - 6⊗6=0 (even) - 8⊗8=1 (odd) - Odd results for n=4 and n=8. **Final answers:** - Addition table and multiplication table as above. - (4 ⊕ 6) ⊕ (2 ⊗ 4) = 0 - n = 2 - Truth set = {4, 8}