Answer
check in solution
Work Step by Step
The compound proposition that asserts that each of the nine 3 × 3 blocks of a 9 × 9 Sudoku puzzle contains every number is composed of several smaller propositions. Here are the steps to construct this compound proposition:
Step 1: Define the variables
First, we need to define the variables that we will use in our proposition. We can use the variables p(i,j,k) to represent the statement that the number k appears in the cell in the i-th row and the j-th column of the Sudoku puzzle. We can also use the variables B(m,n,k) to represent the statement that the number k appears in the m-th 3 × 3 block of the Sudoku puzzle.
Step 2: Express the smaller propositions
Next, we need to express the smaller propositions that make up the compound proposition. These smaller propositions state that every number appears in each of the nine 3 × 3 blocks of the Sudoku puzzle. We can express these propositions as follows:
For each k from 1 to 9, there exists an m from 1 to 9 such that B(m,n,k) is true for every n from 1 to 9. This proposition states that for each number k, there is at least one 3 × 3 block in which it appears in every cell.
For each k from 1 to 9 and for each m from 1 to 9, there exists an i and a j between 1 and 3 such that p(3m+i,3n+j,k) is true for every n from 1 to 3. This proposition states that for each number k and each 3 × 3 block m, there is at least one cell in each row and each column of the block that contains k.
Step 3: Combine the smaller propositions
Finally, we can combine the smaller propositions using logical operators to form the compound proposition that asserts that each of the nine 3 × 3 blocks of a 9 × 9 Sudoku puzzle contains every number. We can express this proposition as follows:
For each k from 1 to 9, there exists an m from 1 to 9 such that B(m,n,k) is true for every n from 1 to 9, and for each k from 1 to 9 and for each m from 1 to 9, there exists an i and a j between 1 and 3 such that p(3m+i,3n+j,k) is true for every n from 1 to 3.
This proposition asserts that for each number k, there is at least one 3 × 3 block in which it appears in every cell, and for each number k and each 3 × 3 block m, there is at least one cell in each row and each column of the block that contains k. This means that each of the nine 3 × 3 blocks of the Sudoku puzzle contains every number.
