The negation of the statement "$2 + 3 = 5$ and $8 < 10$" is:

The contrapositive of "If $x$ and $y$ are integers such that $x y$ is odd,then both $x$ and $y$ are odd" is

The correct logical equivalences from the following are:
$(I)$ $p \to (q \to r) \equiv (p \land q) \to r$
$(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$
$(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$
$(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$

The number of switches in the alternative equivalent simple circuit for the given circuit is:

If $P$ and $Q$ are two statements,then which of the following compound statements is a tautology?

If the truth value of the compound statement $[(p \leftrightarrow q) \land (q \to r) \land \sim r] \to (p \land \sim q)$ is false, then the truth values of the statement patterns $(p \to q) \leftrightarrow (q \to r)$ and $\sim (p \lor r) \to (q \land p)$ are, respectively ...