Determine the validity of the following arguments using the direc… - Vidyadip

Question

Determine the validity of the following arguments using the direct method of truth table:
$P \rightarrow \sim (Q\ \&\ R)$
$\therefore\ \sim (Q\ \&\ R) \rightarrow P$

✓

Answer

Truth Table:

						Support Statement	The resulting statement
	$1$	$2$	$3$	$4$	$5$	$6$	$7$
	$P$	$Q$	$R$	$Q\ \&\ R$	$\sim (Q\ \&\ R)$	$P \rightarrow \sim (Q\ \&\ R)$	$\sim (Q\ \&\ R) \rightarrow P$
$1$	$T$	$T$	$T$	$T$	$F$	$F$	$T$
$2$	$T$	$T$	$F$	$F$	$T$	$T$	$T$
$3$	$T$	$F$	$T$	$F$	$T$	$T$	$T$
$4$	$T$	$F$	$F$	$F$	$T$	$T$	$T$
$5$	$F$	$T$	$T$	$T$	$F$	$T$	$T$
$6$	$F$	$T$	$F$	$F$	$T$	$T^*$	$F^*$
$7$	$F$	$F$	$T$	$F$	$T$	$T^*$	$F^*$
$8$	$F$	$F$	$F$	$F$	$T$	$T^*$	$F^*$
				$2,3(\&)$	$4(\sim )$	$1, 2(\rightarrow)$	$5,1(\rightarrow)$

Judgment of the validity of the argument: A total of seven columns are formed in the above truth table. In which the column no. $6th$ base statement and column no. $7$ is the introduction of the result statement. Flowers of the table of truth out of eight rows. In $2, 3, 4, 5, 6, 7$ and $8$ the base statement truth is $‘T’.$ But of the row. The result statement in $6, 7$ and $8$ is false $‘F’.$ Hence this argument is disproportionate.

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