(1) $(\sim p)\land(p\lor q)\to q$

(2) $(q\to p)\ \lor\sim(p\to q)$

(3) $(\sim q)\lor(p\land q)\to q$

(4) $(p\to q)\land(q\to p)$

**Ans. (1)** $(\sim p)\land(p\lor q)\to q$

**Sol. (i)** $\sim p\land(p\lor q)\to q$

$(\sim p\land p)\lor(\sim p\land q)\to q$

$C\lor(\sim p\land q)\to q$

$(\sim p\land q)\to q$

$\sim(\sim p\land q)\lor q$

$=(p\lor\sim q)\lor q=p\lor t=t$

**(iii)** $(\sim q)\lor(p\land q)\to q$

use $\sim(p\to q)=p\land\sim q\implies p\to q = \sim p\lor q$

$=(\sim q\lor p)\land(\sim q\lor q)\to q$

$=(\sim q\lor p)\to q$

$=(q\land\sim p)\to q$

$=q$

**(ii)** and **(iv)**

p | q | $p\to q$ | $q\to p$ | $\sim(p\to q)$ | $(p\to q)\land(q\to p)$ | $(q\to p)\ \lor\sim(p\to q)$ |

T | T | T | T | F | T | T |

T | F | F | T | T | F | T |

F | T | T | F | F | F | F |

F | F | T | T | T | T | T |