Question

In Exercise 5-10 assume that \(A\) is a subset of some underlying universal set \(U\).

Prove the complementation law in Table 1 by showing that \(\overline{\overline A} = A\).

Short answer

Thus, it is proved that \(\overline{\overline A} = A\)..

Step-by-step solution

Showing that \(A \subseteq \overline{\overline A} \).

Let for arbitrary \(x\), If \(x \in A\) then \(x \notin \overline A \) so, \(x \in \overline{\overline A} \).

Thus, write as follows:

\(A \subseteq \overline{\overline A} \)

Hence, the set is \(A \subseteq \overline{\overline A} \).

Showing that \(\overline{\overline A}  \subseteq A\).

If \(x \in \overline{\overline A} \) then \(x \notin \overline A \) so,\(x \in A\).

Thus, write as follows:

\(\overline{\overline A} \subseteq A\)

Hence, the set is \(\overline{\overline A} \subseteq A\).

When \(A \subseteq \overline{\overline A} \) and \(\overline{\overline A} \subseteq A\) then, \(\overline{\overline A} = A\).

Therefore, it is proved that \(\overline{\overline A} = A\).