Question

The only statement among the followings that is a tautology is

• A. $p\wedge (p\vee q)$
• B. $p\vee (p\wedge q)$
• C. $[p\wedge (p\to q\left)\right]\to q$
• D. $q\to [p\wedge (p\to q\left)\right]$

Answer 1

The correct option is C $[p\wedge (p\to q\left)\right]\to q$

$p\wedge (p\vee q)$ is $F$ when $p=F$
$p\vee (p\wedge q)$ is $F$ when $p=F$

$[p\wedge (p\to q\left)\right]\to q$
$\equiv [p\wedge (\sim p\vee q\left)\right]\to q$
$\equiv \left[\right(p\wedge \sim p)\vee (p\wedge q\left)\right]\to q$
$\equiv [F\vee (p\wedge q\left)\right]\to q$
$\equiv (p\wedge q)]\to q$
$\equiv \sim (p\wedge q)\vee q$
​​​​​​​$\equiv (\sim p\vee \sim q)\vee q$
​​​​​​​​​​​​​​$\equiv \sim p\vee (q\vee \sim q)$
​​​​​​​​​​​​​​$\equiv \sim p\vee T$
$\equiv T$