a) Suppose p, q and r mean “Kelly is at home, Hannah is at home and Sunny is at home then interpret the meaning of the statement p ∧∼q and (p ∧ q) ∧ r (b) You should pay the fares for tickets only if you plan to visit northern Pakistan in this winter. (Convert the sentence in symbolic notation) (c) If the lift in our office install then one can use the lift (write a converse of the statement in symbolic notation as well as a statement) (d) If we do not make a plan to plant marigold flowers then it is not suitable season for them. (Write the contrapositive statement in symbolic as well as a statement) (e) If it is right angled triangle then Pythagoras rule applies. (Convert the statements in biconditional statement in symbolic as well as a statement)

Solution:

(a) $p\land \lnot q:\text{Kelly is at home and Hannah is not at home}$

$(p\land q)\land r: \text{Kelly, Hannah and Sunny are at home}$

(b) Define $p: \text{You should pay the fares for the tickets}$

$q:\text{You plan to visit Northern Pakistan in winter}$

Then symbolically,

$p\rightarrow q$

(c) Define $p: \text{The lift in our office is installed }$

$q:\text{One can use the lift}$

Converse: If we can use the lift , then the lift in our office is installed

Symbolically: $q\rightarrow p$

(d) Define $p:\text{We do not make a plan to plant marigold flowers}$

$q:\text{It is nott suitable season for them}$

Contrapositive : If it is a suitable season, then we make a plan to plant marigold flowers

Symbolically: $q\rightarrow p$

(e) Define $p:\text{It is a right angled triangle}$

$q:\text{Pythagoras rule applies }$

Biconditional : A triangle is a right angled triangle iff it satisfies Pythagoras rule

Symbolically: $p\leftrightarrow q$