Exercise 12.2-Additional Problems - Chapter 12 - Discrete Mathematics - 12th Maths Guide Samacheer Kalvi Solutions - Tamil Medium

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Additional Problems
Question 1.
Show that $((\neg p) \vee(\neg q)) \vee p$ is a tautology.
Solution:
Truth table for $((\neg p) \vee(\neg q)) \vee p$

The last column contains only T. $\therefore((\neg p) \vee(\neg q)) \vee p$ is a tautology
Question 2.
Show that $((\neg \boldsymbol{q}) \wedge \boldsymbol{p}) \wedge \boldsymbol{q}_{\text {is a contradiction. }}$
Solution:
Truth table for $((\neg q) \wedge p) \wedge q$

The last column contains only F. $\therefore((\neg q) \wedge p) \wedge q$ is a contradiction.
Question 3.
Use the truth table to determine whether the statement $((\neg p) \vee q) \vee(p \wedge(\neg q))$ is a tautology.
Solution:
Truth table for $((\neg p) \vee q) \vee(p \wedge(\neg q))$

The last column contains only T. $\therefore$ The given statement is a tautology.
Question 4.
Show that $p \leftrightarrow q \equiv((\neg p) \vee q) \wedge((\neg q) \vee p)$
Solution:
(i) Truth table for $\mathrm{p} \leftrightarrow \mathrm{q}$

$\text { (ii) Truth table for }((\neg p) \vee q) \wedge((\neg q) \vee p)$

The last columns of statements ( $i$ ) and (ii) are identical.
So, $p \leftrightarrow q \equiv((\neg p) \vee q) \wedge((\neg q) \vee p)$
Question 5.
Show that $\neg(p \wedge q) \equiv((\neg p) \vee(\neg q))$.
Solution:

$\text { (ii) Truth table for }((\neg p) \vee(\neg q))$

The last columns of statements (i) and (ii) are identical.
$
\text { So } \neg(p \wedge q) \equiv((\neg p) \vee(\neg q))
$