Write the following propositions using the propositional variables

p and $q,$ along with logical connectives $({?}^{{?}^{?}},v,,-$ : $(5$ marks @ $1$ marks each$)$

a$)$ If it rains, then the ground is wet.

b$)$ It is not the case that both John and Mary are at home.

c$)$ Either the light is on or the switch is faulty, but not both.

d$)$ If the train arrives late, then either I will be late or I will miss the meeting.

e$)$ Both the network is down and the server is not responding.

Let $p,q$ and $r$ be the propositions $(6$ marks @ $1$ mark for each$)$

p : It's your birthday

q: There will be cake

$r$ : We are going to celebrate the birthday.