Question

Simplify the following statements:
~(~p^q)

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\: \sim( \sim \: p \: \land \: q)$

We know,

$\boxed{ \tt{ \: \sim (\sim \: p) \: = \: p \: }}$

So,

$\red{\bf\implies \:\boxed{ \tt{ \:\: \sim( \sim \: p \: \land \: q) \: = \: p \: \lor \: \sim \: q}}}$

Justification :-

Consider,

$\rm :\longmapsto\: \sim( \sim \: p \: \land \: q)$

Truth table are as follow :-

$\begin{gathered}\boxed{\begin{array}{c|c|c|c|c} \bf p & \bf q& \bf \sim \: p& \bf \sim p\: \land \: q& \bf \sim( \sim \: p \land \: q)\\ \frac{\qquad}{} & \frac{\qquad}{}& \frac{\qquad}{} & \frac{\qquad}{}& \frac{\qquad}{}\\ \sf T & \sf T & \sf F& \sf F& \sf T\\ \\\sf T & \sf F & \sf F& \sf F& \sf T\\ \\\sf F & \sf T & \sf T& \sf T& \sf F\\ \\\sf F & \sf F & \sf T& \sf F& \sf T \end{array}} \\ \end{gathered}$

Now, Consider

$\rm :\longmapsto\:p \: \lor \: \sim \: q$

Truth table are as follow :-

$\begin{gathered}\boxed{\begin{array}{c|c|c|c} \bf p & \bf q& \bf \sim \: q& \bf p \: \lor \: \sim q\\ \frac{\qquad}{} & \frac{\qquad}{}& \frac{\qquad}{} & \frac{\qquad}{}\\ \sf T & \sf T & \sf F& \sf T\\ \\\sf T & \sf F & \sf T& \sf T\\ \\\sf F & \sf T & \sf F& \sf F\\ \\\sf F & \sf F & \sf T& \sf T \end{array}} \\ \end{gathered}$

Hence,

$\red{\bf\implies \:\boxed{ \tt{ \:\: \sim( \sim \: p \: \land \: q) \: = \: p \: \lor \: \sim \: q}}}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c|c|c|c} \bf p & \bf q& \bf p \: \lor \: q& \bf q \: \land \: p& \bf p \: \to \: q\\ \frac{\qquad}{} & \frac{\qquad}{}& \frac{\qquad}{} & \frac{\qquad}{}& \frac{\qquad}{}\\ \sf T & \sf T & \sf T& \sf T& \sf T\\ \\\sf T & \sf F & \sf T& \sf F& \sf F\\ \\\sf F & \sf T & \sf T& \sf F& \sf T\\ \\\sf F & \sf F & \sf F& \sf F& \sf T \end{array}} \\ \end{gathered}$

Answer 2

Step-by-step explanation:

Since columns corresponding to ¬(p∨q) and (¬p∧¬q) match, the propositions are logically equivalent. ... Since columns corresponding to p∨(q∧r) and (p∨q)∧(p∨r) match, the propositions are logically equivalent. This particular equivalence is known as the Distributive Law.