This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here is the solution to question 3: a) To construct a table for multiplication modulo 11 on the replacement set T = \1, 5, 9, 10\, we multiply each pair of elements from T and find the remainder when divided by 11. Step 1: Calculate each product modulo 11. 1 1 = (1 × 1) 11 = 1 1 5 = (1 × 5) 11 = 5 1 9 = (1 × 9) 11 = 9 1 10 = (1 × 10) 11 = 10 5 1 = (5 × 1) 11 = 5 5 5 = (5 × 5) 11 = 25 11 = 3 5 9 = (5 × 9) 11 = 45 11 = 1 5 10 = (5 × 10) 11 = 50 11 = 6 9 1 = (9 × 1) 11 = 9 9 5 = (9 × 5) 11 = 45 11 = 1 9 9 = (9 × 9) 11 = 81 11 = 4 9 10 = (9 × 10) 11 = 90 11 = 2 10 1 = (10 × 1) 11 = 10 10 5 = (10 × 5) 11 = 50 11 = 6 10 9 = (10 × 9) 11 = 90 11 = 2 10 10 = (10 × 10) 11 = 100 11 = 1 Step 2: Construct the multiplication table. |c|c|c|c|c| & 1 & 5 & 9 & 10 \\ 1 & 1 & 5 & 9 & 10 \\ 5 & 5 & 3 & 1 & 6 \\ 9 & 9 & 1 & 4 & 2 \\ 10 & 10 & 6 & 2 & 1 \\ b) Using the table from (a), we find the truth set for t T = \1, 5, 9, 10\. i) 10 t = 2 Step 3: Find t such that 10 t = 2. Looking at the row for 10 in the table, we find that 10 9 = 2. The truth set is \9\. ii) t t = 4 Step 4: Find t such that t t = 4. We check each element in T: For t=1, 1 1 = 1 4. For t=5, 5 5 = 3 4. For t=9, 9 9 = 4. This matches. For t=10, 10 10 = 1 4. The truth set is \9\. c) To evaluate (5 9) 10 from the table in (a). Step 5: Evaluate the expression. First, find 5 9 from the table: 5 9 = 1. Next, substitute this value back into the expression: (5 9) 10 = 1 10. From the table, 1 10 = 10. The value is 10. Two more to go! Send me the next one 📸