Question

Explain the steps in the construction of the compound proposition given in the text that asserts that each of the 3x3nineblocks of a9x9Sudoku 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.The compound proposition given in the text asserts that each of the nine 3x3 blocks of a 9x9 Sudoku puzzle contains every number is ${{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{r}\mathbf{=}\mathbf{0}}{{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{s}\mathbf{=}\mathbf{0}}{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{3}\mathbf{r}\mathbf{+}\mathbf{i}\mathbf{,}\mathbf{3}\mathbf{s}\mathbf{+}\mathbf{j}\mathbf{,}\mathbf{n}\mathbf{)}$.

Step-by-step solution

Definition of Sudoku puzzle

Sudoku is a logic puzzle. Sudoku puzzles are made up of 81 cells that are organized into nine columns, rows, and regions.

Construction of a compound proposition given in the text that asserts that each of the nine  3x3 blocks of a 9x9 Sudoku puzzle contains every number.

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

Suppose (r,s) block for the integer $\mathbf{0}\mathbf{\le }\mathit{r}\mathbf{\le }\mathbf{2}$and $\mathbf{0}\mathbf{\le }\mathit{s}\mathbf{\le }\mathbf{2}$.

This indicates the block in the rows3r+1, 3r+2, 3r+3and the columns of3s+1, 3s+2, 3s+3of the Sudoku.

Here, ${{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathit{P}\mathbf{(}\mathbf{3}\mathit{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}\mathbf{)}$has to be true for every integernfrom1to9.

When${{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathit{P}\mathbf{(}\mathbf{3}\mathit{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}\mathbf{)}$ has to be true, localid="1668249341503" ${{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathit{P}\mathbf{(}\mathbf{3}\mathit{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}\mathbf{)}$needs to be true for every block (r,s).

However, ${{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathit{P}\mathbf{(}\mathbf{3}\mathit{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}\mathbf{)}$ has to be true for every block (r,s) then ${{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{r}\mathbf{=}\mathbf{0}}{{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{s}\mathbf{=}\mathbf{0}}{\mathbf{\wedge }\mathbf{9}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathbf{P}\mathbf{(}\mathbf{3}\mathbf{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}{\mathbf{)}}_{}$needs to be true.

Therefore, the compound proposition given in the text asserts that each of the nine 3x3blocks of a 9x9Sudoku puzzle contains every number is ${{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{r}\mathbf{=}\mathbf{0}}{{\mathbf{\wedge }}^{\mathbf{2}}}_{\mathbf{s}\mathbf{=}\mathbf{0}}{{\mathbf{\wedge }}^{\mathbf{9}}}_{\mathbf{n}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{i}\mathbf{=}\mathbf{1}}{{\mathbf{\vee }}^{\mathbf{3}}}_{\mathbf{j}\mathbf{=}\mathbf{1}}\mathit{P}\mathbf{(}\mathbf{3}\mathit{r}\mathbf{+}\mathit{i}\mathbf{,}\mathbf{3}\mathit{s}\mathbf{+}\mathit{j}\mathbf{,}\mathit{n}\mathbf{)}$.