Question

Explain the steps in the construction of the compound proposition given in the text that asserts that every column of a 9×9 Sudoku puzzle contains every number.

Short answer

Suppose that p(i,j,n) indicates the cell in row i and column j and also has a value of n .A compound proposition given in the text that asserts that every column of a 9x9Sudoku puzzle contains every number is ${{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$.

Step-by-step solution

Definition of Sudoku puzzle

Sudoku is a logic puzzle. Sudoku puzzles are made up of cells that are organized into nine columns, rows, and regions.

Construction of a compound proposition given in the text that asserts that every column of a 9x9 Sudoku puzzle contains every number.

Let p(i,j,n) be the cell in a row in row iand column jand also has a value of n.

Every column must contain every number between1 and 9.

Thus, there are 9 rows and 9 columns.

Here, ${{\mathbf{\vee }}^{\mathbf{9}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$has to be true for every column j and some row.

Also, ${{\mathbf{\vee }}^{\mathbf{9}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$ needs to be true for every integer from 1 to 9. So, role="math" localid="1668257130533" ${{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{\vee }}^{\mathbf{9}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$ has to be true for every column j.

When ${{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{\mathbf{\vee }}^{\mathbf{9}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$ needs to be true for every column j,role="math" localid="1668257285957" ${{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathit{i}\mathbf{=}\mathbf{1}}{{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathit{j}\mathbf{=}\mathbf{1}}{\mathbf{\vee }}^{\mathbf{9}}}_{\mathit{n}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$ needs to be true.

Therefore, a compound proposition given in the text asserts that every column of a 9x9 Sudoku puzzle contains every number is ${{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathit{i}\mathbf{=}\mathbf{1}}{{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathit{j}\mathbf{=}\mathbf{1}}{\mathbf{\vee }}^{\mathbf{9}}}_{\mathit{n}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$.