Method

Making a Truth Table

There are three steps to follow to make a truth table for a compound statement.

Determine the number of columns of the truth table.
Determine the number of rows of the table.
Fill in the cells of the table.

Let

p

and

q

represent two different logical statements. Consider the following compound statement.

(\neg p \lor q) \land \neg q

By making a truth table it can be determined for what values of

p

and

q

this statement is true and for what values it is false.

Determine the Number of Columns

The number of columns of a truth table depends on the complexity of the statement. To determine this number, the compound statement must be broken down into simpler statements until the simplest statements are obtained. In this case, the compound statement is broken down until $p$ and $q$ are obtained.

The columns are formed by ordering each unique step of simplification from simplest to most complex. The last column of a truth table is always the original statement.

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
$?$	$?$	$?$	$?$	$?$	$?$

In this case, the truth table has six columns.

Determine the Number of Rows

A truth table has as many rows as there are combinations of the truth values of the variables, which are also the simplest statements of the complex statement. In this case, the complex statement was broken down into two simplest statements.

Simplest Statements p and q

Each of these statements can be either true (T) or false (F). Therefore, since there are four possible combinations for these truth values, the table will have four rows.

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
T	T	$?$	$?$	$?$	$?$
T	F	$?$	$?$	$?$	$?$
F	T	$?$	$?$	$?$	$?$
F	F	$?$	$?$	$?$	$?$

The general rule is that a truth table has $2^{n}$ rows, where $n$ is the number of variables.

Fill in the Cells

The last step in making a truth table is to fill in the remaining empty cells, column by column. The first column that will be filled in is the negation $(\neg)$ of $p .$ A negation has the opposite truth value than the original statement.

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
$T$	T	$F$	$?$	$?$	$?$
$T$	F	$F$	$?$	$?$	$?$
$F$	T	$T$	$?$	$?$	$?$
$F$	F	$T$	$?$	$?$	$?$

Similarly, the fourth column — the negation of $q$ — is completed.

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
T	$T$	F	$F$	$?$	$?$
T	$F$	F	$T$	$?$	$?$
F	$T$	T	$F$	$?$	$?$
F	$F$	T	$T$	$?$	$?$

The next column is the disjunction $(\lor)$ of $\neg p$ and $q .$ A disjunction is $true$ unless both statements are $false .$

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
T	$T$	$F$	F	$T$	$?$
T	$F$	$F$	T	$F$	$?$
F	$T$	$T$	F	$T$	$?$
F	$F$	$T$	T	$T$	$?$

Finally, the last column is the conjunction $(\land)$ of $(\neg p \lor q)$ and $\neg q .$ A conjunction is $false$ unless both statements are $true .$

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
T	T	F	$F$	$T$	$F$
T	F	F	$T$	$F$	$F$
F	T	T	$F$	$T$	$F$
F	F	T	$T$	$T$	$T$

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$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
$T$	T	$F$	$?$	$?$	$?$
$T$	F	$F$	$?$	$?$	$?$
$F$	T	$T$	$?$	$?$	$?$
$F$	F	$T$	$?$	$?$	$?$

$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
$T$	T	$F$	$?$	$?$	$?$
$T$	F	$F$	$?$	$?$	$?$
$F$	T	$T$	$?$	$?$	$?$
$F$	F	$T$	$?$	$?$	$?$

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$p$	$q$	$\neg p$	$\neg q$	$\neg p \lor q$	$(\neg p \lor q) \land \neg q$
$T$	T	$F$	$?$	$?$	$?$
$T$	F	$F$	$?$	$?$	$?$
$F$	T	$T$	$?$	$?$	$?$
$F$	F	$T$	$?$	$?$	$?$