Question

Assignment 2 2. Which of the following formulas are tautologies? Justify your answers. (a) (P+ (PVO))

Answer 1

If a statement is a tautology, then all the values in the final column of its truth table will be true. i.e. The statement will be true for all the cases.

So construct a truth table for each of the statements to check whether they are tautology.

a. Consider

(P + (PV)

Now Construct a truth table for the formula :

(P + (PV)

		$\left ( P\vee Q \right )$	(P + (PV)
0	0	0	1
0	1	1	1
1	0	1	1
1	1	1	1

Notice that the given formula is true for all the possible cases.

(P + (PV)

hence is a tautology.

(P + (PV)

.

(b). Consider .

(PARQ) + R)

Now Construct a truth table for the formula :

(PARQ) + R)

			$\sim Q$	$\left (P\wedge \sim Q \right )$	$\left (P\wedge \sim Q \right )\rightarrow R$
0	0	0	1	0	1
0	0	1	1	0	1
0	1	0	0	0	1
0	1	1	0	0	1
1	0	0	1	1	0
1	0	1	1	1	1
1	1	0	0	0	1
1	1	1	0	0	1

From the truth table, note that the formula
(PARQ) + R)
is not true for all cases. It is false when P is true and Q and R are both false. Hence the given formula
(PARQ) + R)
is not a tautology.