$ \sim[(\sim p) \wedge q] $ is logically equivalent to

The simplified switching circuit for the following circuit is (Assume the circuit consists of two switches $S_1$ and $S_2$ in parallel, connected in series with a switch $S_3$):

The logical expression $[p \wedge (q \vee r)] \vee [\sim r \wedge \sim q \wedge p]$ is equivalent to

Consider the following statements:
$P :$ Suman is brilliant
$Q :$ Suman is rich
$R :$ Suman is honest
The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as:

Which of the following is logically equivalent to $(p \wedge q)$?

Consider the statement: "For an integer $n$,if $n^{3}-1$ is even,then $n$ is odd." The contrapositive statement of this statement is