Question

~(p Λ q) is logically equivalen to ______
Select one:
a. p Λ ~q
b. ~ p V ~q
c. ~ p Λ q
d. ~ p Λ ~q

Answer 1

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

Consider

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

Its means option (a) is not correct.

Now, Consider

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

It means, option (b) is correct.

Now, Consider

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

It means, Option (c) is not correct

Now, Consider

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

It means, Option (d) is not correct.

$\bf\implies \: \sim\: (p \: \land \: q) \: \equiv \: \sim \: p \: \lor \: \sim \: q$

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More to Know

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

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