question In Detail . (5 Marks)

Question

Explain the method of determining the value of truth by truth table.

Vidyadip · Accepted Answer

Columns and rows have to be formed to form a truth table.

The column from top to bottom of the table of truth is called a column; While the horizontal box from left to right is called a row.
The rules regarding the number of columns and rows are as follows:
$1.$ The rule regarding the formation of the first column in the fact table: In the fact table for any statement, the first column $($main column$)$ has to be formed for every simple statement in the statement.
The simpler the statement, the simpler the statement. For example, $A \leftrightarrow (A\leftrightarrow B) $ forms the first column for statement $A$ and another for statement $B$ in the truth table for this complex joint statement.
Even if a statement occurs twice, its pillar is kept the same.
$2.$ The rule regarding the formation of rows in a fact sheet: The following formula is used to determine the number of rows in any joint statement:
Number of rows $- 2n (n =$ full number of starting column$)$
$(i) A \ v\ \sim A:$ There is only one simple statement A in this complex joint statement. Thus the pillar will be one. The verification table of this statement will have two lines $21 = 2 \times 1 = 2 m.$
$(ii) L\ (L \rightarrow M):$ In this complex compound statement, $L$ and $M$ are two simple statements. This will lead to two pillars. The verification table of this statement will have four lines $22 = 2 \times 2 = 4 m.$
$(iii) (P\ v\ Q) \leftrightarrow R: P, Q$ and $R$ are the three simple statements in this complex joint statement. So there will be three pillars.
In the verification of this statement $23 = 2 \times 2\times 2 = 8 M$ will come in eight rows.
Thus, according to this rule, there are two rows in a table with one column, four rows in a table with two pillars and eight rows in a table with three pillars.
$3.$ The rule regarding the arrangement of $‘T’$ and $‘F’$ in each pillar:
$(1)$ The rule of arrangement for truthfulness in a column:

	$p$	Truthfulness
$1$	$T$
$2$	$F$

$(2)$ Rule of arrangement for truthfulness in two pillars:

	$p$	$p$	Truthfulness
$1$	$T$	$T$
$2$	$T$	$F$
$3$	$F$	$T$
$4$	$F$	$F$

$(3)$ Rule of arrangement for truthfulness in three pillars:

	$p$	$q$	$r$	Truthfulness
$1$	$T$	$T$	$T$
$2$	$T$	$T$	$F$
$3$	$T$	$F$	$T$
$4$	$T$	$F$	$F$
$5$	$F$	$T$	$T$
$6$	$F$	$T$	$F$
$7$	$F$	$F$	$T$
$8$	$F$	$F$	$F$

The following are significant points regarding the composition of the fact sheet as above:

The first half of the total row is written $‘T’$ under the first column and the remaining half is written $‘F’.$
Under the second column, one-fourth of the total number of strings is written, first ‘T’ and then $‘F’.$
Under the third column, first $‘T’ $ and then $‘F’$ are written alternately.
In this way, after constructing the column and the necklace, the truthfulness of the prohibition, set, alternative, conditional and bi$-$conditional statement which is $-$ is done according to the rule of the joint statement and is indicated in the row under the right column.

$4.$ Rule relating to the formation of columns other than pillars in the fact$-$table $- ($secondary columns$):$ When constructing a fact-table for any complex joint statement, first the pillars have to be made and then other columns or secondary columns have to be formed.

In any complex compound statement, as many times as one or different logical factors are used, other columns or secondary columns have to be formed in the fact sheet for that statement.
Thus, the number of other columns or secondary columns is formed as many times as the logical factors occur in a complex composite statement.

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