Algebra of Statements - Practice Questions & MCQ

Idempotent Law
1. $p \vee p \equiv p$
2. $p \wedge p \equiv p$

$
\begin{array}{|c|c|c|}
\hline p & p \vee p & p \wedge p \\
\hline \hline \mathrm{~T} & \mathrm{~T} & \mathrm{~T} \\
\hline \mathrm{~F} & \mathrm{~F} & \mathrm{~F} \\
\hline
\end{array}
$

Associative Law
1. $(p \vee q) \vee r \equiv p \vee(q \vee r)$
2. $(p \wedge q) \wedge r \equiv p \wedge(q \wedge r)$

Distributive Law
1. $p \wedge(q \vee r) \equiv(p \wedge q) \vee(p \wedge r) \mid$
2. $p \vee(q \wedge r) \equiv(p \vee q) \wedge(p \vee r)$

Commutative Law
1. $p \vee q \equiv q \vee p$
2. $p \wedge q \equiv q \wedge p$

Identity Law
1. $p \wedge T \equiv p$
2. $p \wedge F \equiv F$
3. $\mathrm{p} \vee T \equiv T$
4. $p \vee F \equiv p$

$
\begin{array}{|c|c|c|c|c|c|c|}
\hline p & T & F & p \wedge T & p \wedge F & p \vee T & p \vee F \\
\hline \hline \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} \\
\hline \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} \\
\hline
\end{array}
$

Complement Law
5. $p \vee \sim p \equiv T$
6. $p \wedge \sim p \equiv F$
7. $\sim(\sim p) \equiv p$
8. $\sim T \equiv F$
9. $\sim \mathrm{F} \equiv \mathrm{T}$

$
\begin{array}{|c|c|c|c|c|c|}
\hline p & \sim p & p \vee \sim p & T & p \wedge \sim p & F \\
\hline \hline \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline
\end{array}
$

De-Morgan's Law
1. $\sim(p \vee q) \equiv \sim p \wedge \sim q$
2. $\sim(p \wedge q) \equiv \sim p \vee \sim q$

Truth table for $\sim(p \vee q)$ and $\sim p \wedge \sim q$

$
\begin{array}{|c|c|c|c|c|c|c|}
\hline p & q & \sim p & \sim q & p \vee q & \sim(p \vee q) & \sim p \wedge \sim q \\
\hline \hline \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} \\
\hline
\end{array}
$

$
\begin{array}{|c|c|c|c|c|c|c|}
\hline p & q & \sim p & \sim q & p \wedge q & \sim(p \wedge q) & \sim p \vee \sim q \\
\hline \hline \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} \\
\hline \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} \\
\hline \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} \\
\hline \mathrm{~F} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} \\
\hline
\end{array}
$