Step 1

(i)

Let,

p:Karen is studying computing.

q:Ketsi is studying mathematics.

\(\displaystyle\sim{q}\):Ketsi is now studying mathematics.

Then,\(\displaystyle{\left({p}\wedge!{q}\right)}\vee{q}\):Either Karen is studying computing and Ketsi is not mathematics, or Ketsi is studying mathematics.

Here, principal connective is "vv"(i.e."or").

Step 2

(ii)

Let,

p: It is sunny.

q: I will carry an umberella.

Then, \(\displaystyle!{\left({p}\rightarrow{q}\right)}\): It is not the case that if it is sunny then i will carry an umbrella.

Here, principal connective is "\(\displaystyle\sim\)"(i.e."not").

(i)

Let,

p:Karen is studying computing.

q:Ketsi is studying mathematics.

\(\displaystyle\sim{q}\):Ketsi is now studying mathematics.

Then,\(\displaystyle{\left({p}\wedge!{q}\right)}\vee{q}\):Either Karen is studying computing and Ketsi is not mathematics, or Ketsi is studying mathematics.

Here, principal connective is "vv"(i.e."or").

Step 2

(ii)

Let,

p: It is sunny.

q: I will carry an umberella.

Then, \(\displaystyle!{\left({p}\rightarrow{q}\right)}\): It is not the case that if it is sunny then i will carry an umbrella.

Here, principal connective is "\(\displaystyle\sim\)"(i.e."not").