8. Five mark example problems IV

If \(U = \{x: x \in Z, -2 \leq x \leq 10\}\), \(A = \{x: x = 2p + 1, p \in Z, -1 \leq p \leq 4\}\), \(B = \{x: x = 3p + 1, q \in Z, -1 \leq q < 4\}\), verify De Morgan's laws for complementation.

 

Proof:

 

Law (i): Let us prove that \((A \cup B)' = A' \cap B'\)

 

\(A \cup B =\) $\left\{i,i,i,i,i,i,i,i,i\right\}$

 

\((A \cup B)' =\) $\left\{i,i,i,i\right\}$ ---- (\(1\))

 

\(A' =\) $\left\{i,i,i,i,i,i,i\right\}$

 

\(B' =\) $\left\{i,i,i,i,i,i,i,i\right\}$

 

\(A' \cap B' =\) $\left\{i,i,i,i\right\}$ ---- (\(2\))

 

From equations (\(1\)) and (\(2\)), \((A \cup B)' = A' \cap B'\).

 

Hence, we proved.

 

Law (ii): Let us prove that \((A \cap B)' = A' \cup B'\)

 

\(A \cap B =\) $\left\{i,i\right\}$

 

\((A \cap B)' =\) $\left\{i,i,i,i,i,i,i,i,i,i,i\right\}$ ---- (\(3\))

 

\(A' \cup B' =\) $\left\{i,i,i,i,i,i,i,i,i,i,i\right\}$ ---- (\(4\))

 

From equations (\(3\)) and (\(4\)), \((A \cap B)' = A' \cup B'\).

 

Hence, we proved.

 

(Note: Enter the numbers in ascending order.)