Contrapositive o the statement : 'If a function f is differentiable at a, then it is also continuous at a', is :

• If a function f is not continuous at a, then it is not differentiable at a.

• If a function f is continuous at a, then it is differentiable at a.

• If a function f is not continuous at a. then it is differentiable at a.

• If a function f is continuous at a, then it is not differentiable at a.

The negation of the Boolean expression $\mathrm{x} \leftrightarrow ~ \mathrm{y}$ is equivalent to : 

• $\left(~ \mathrm{x} \wedge \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge ~ \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge ~ \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge ~ \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge \mathrm{y}\right) \wedge \left(~ \mathrm{x} \vee ~ \mathrm{y}\right)$

The statement

$\left(\mathrm{p} \to \left(\mathrm{q} \to \mathrm{p}\right)\right) \to \left(\mathrm{p}\to \left(\mathrm{p} \vee \mathrm{q}\right)\right) \mathrm{is}$

• equivalent to $\left(\mathrm{p} \vee \mathrm{q}\right) \wedge \left(~ \mathrm{p}\right)$

• a contradiction

• a tautology

• equivalent to $\left(\mathrm{p} \wedge \mathrm{q}\right) \vee \left(~ \mathrm{q}\right)$

44.

The negation of the Boolean expression p v (~ p $\wedge$ q) is equivalent to :

• $\mathrm{p} \wedge ~ \mathrm{q}$

• $~ \mathrm{p} \vee \mathrm{q}$

• $~ \mathrm{p} \wedge ~ \mathrm{q}$

• $~ \mathrm{p} \vee ~ \mathrm{q}$

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{statement}: “\mathrm{For} \mathrm{an} \mathrm{integer} \mathrm{n}, \mathrm{if} {\mathrm{n}}^3 – 1 \mathrm{is} \mathrm{even}, \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{odd}”. \mathrm{The} \\ \mathrm{contrapositive} \mathrm{statement} \mathrm{of} \mathrm{this} \mathrm{statement} \mathrm{is}: \end{gathered}$$

• For an integer n, if n is even, then n³ – 1 is odd

• For an integer n, if n is even, then n³ – 1 is even.

• For an integer n, if n³ – 1 is not even, then n is not odd

• For an integer n, if n is odd, then n³ – 1 is even.