Truth Table - Practice Questions & MCQ

Truth Value of a Statement

As we know a statement is either true or false. The truth or falsity of a statement is called truth value.

If the statement is true, then the truth value is “T” 

If the statement is false, then the truth value is “F” 

Truth Table

A table indicating the truth value of one or more statements is called a truth table.

The truth table of one statement ‘p’ is 

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

The truth table for two statements ‘p’ and ‘q’ is

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

In the case of n statements, there are 2ⁿ distinct possible arrangements of truth values of statements.

Truth Table for Negation of a Statement 

The truth value of the negation of a statement is always opposite to the truth value of the original statement.

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

Truth Table of Conjunction and Disjunction:

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

Negation of a Negation

Negation of negation of a statement is the statement itself.  Equivalently, we write: ~ ( ~ p) = p

Truth Table

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

Truth Table for Conditional Statement:

A Conditional Statement is false only when p is true and q is false. In all other cases, this is true.

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

Truth Table for Biconditional Statements:

A biconditional statement is true when both p and q are true or when both p and q are false

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