GATE CSE 1997 | Question: 3.2

Question

Which of the following propositions is a tautology?

• A. $(p \vee q) \rightarrow p$
• B. $p \vee (q \rightarrow p)$
• C. $p \vee (p \rightarrow q)$
• D. $p \rightarrow (p \rightarrow q)$

Answer 1

$A. \ \ (p \vee q) \rightarrow p$
$\quad \quad \equiv \ \ \neg(p \vee q) \vee p$
$\quad \quad \equiv \ \ (\neg p \wedge \neg q) \vee p$
$\quad \quad \equiv \ \ (\neg p \vee p)\wedge(\neg q \vee p)$
$\quad \quad \equiv (p \vee\neg q)$

$B. \ \ p \vee (q \rightarrow p)$
$\quad \quad \equiv \ \ p \vee (\neg q \vee p)$
$\quad \quad \equiv \ \ (p \vee p)\vee\neg q$
$\quad \quad \equiv p \vee\neg q$

$C. \ \ p \vee (p \rightarrow q)$
$\quad \quad \equiv \ \ p \vee (\neg p \vee q)$
$\quad \quad \equiv \ \ ( p \vee \neg p) \vee q$
$\quad \quad \equiv \ \ T \vee q$
$\quad \quad \equiv \ \ T$

$D. \ \ p \rightarrow (p \rightarrow q)$
$\quad \quad \equiv \ \ p \rightarrow (\neg p \vee q)$
$\quad \quad \equiv \ \ \neg p \vee(\neg p \vee q)$
$\quad \quad \equiv \ \ (\neg p \vee\neg p) \vee q$
$\quad \quad \equiv \ \ \neg p \vee q$

Hence, Option(C) $p \vee (p \rightarrow q)$.

Answer 2

C) P OR (P -->Q) = P OR (NOT P OR Q)

                          =(P OR NOT P) OR Q                  Associativity rule

                          =T OR Q                                     

                          =T

Comments

Since only p and q are there, we can also give 1/0 to them and see that all other options can give F.

Answer 3

[c]

p v (p->q) can be written as

p+(p' + q)

p+p'+q

1+q

1

hence  it's a tautology