The statement $\left(\mathrm{p} \Rightarrow \mathrm{q}\right) \iff \left(~ \mathrm{p} \wedge \mathrm{q}\right)$ is

• tautology

• contradiction

• Neither (a) nor (b)

• None of these

The negation of $\left(~ \mathrm{p} \wedge \mathrm{q}\right) \vee \left(\mathrm{p} \wedge ~ \mathrm{q}\right)$ is

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

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

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

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

If a₁, a₂ and a₃ be any positive real numbers, then which of the following statement is not true?

• 3a₁a₂a₃ $\le {{\mathrm{a}}_1}^3 + {{\mathrm{a}}_2}^3 + {{\mathrm{a}}_3}^3$

• $\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2} + \frac{{\mathrm{a}}_2}{{\mathrm{a}}_3} + \frac{{\mathrm{a}}_3}{{\mathrm{a}}_1} \ge 3$

• (a₁ + a₂ + a₃)$\left(\frac{1}{{\mathrm{a}}_1} + \frac{1}{{\mathrm{a}}_2} + \frac{1}{{\mathrm{a}}_3}\right) \ge 9$

• (a₁ . a₂ . a₃)$\left(\frac{1}{{\mathrm{a}}_1} + \frac{1}{{\mathrm{a}}_2} + \frac{1}{{\mathrm{a}}_3}\right) \ge 27$

The proposition $~ \left(\mathrm{p} \iff \mathrm{q}\right)$ is equivalent to

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

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

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

• None of the above

If truth values of P be F and q be T. Then, truth value of $~ \left(~ \mathrm{p} \vee \mathrm{q}\right)$ is

• T

• F

• Either T or F

• Neither T or F

Let A be any element in a Boolean algebra B, if a + x = 1 and ax = 0, then

• x = 1

• x = 0

• x = a

• x = a'

Dual of (x + y) . (x + 1) = x + x . y + y is

• (x . y) + (x . 0) = x . (x + y) . y

• (x + y) + (x . 1) = x . (x + y) . y

• (x . y) (x . 0) = x . (x + y) . y

• None of these

Let p and q be two statements. Then, p v q is false, if

• pis false and q is true

• both p and q are false

• both p and q are true

• None of the above

If $\mathrm{p} \Rightarrow \left(~ \mathrm{p} \vee \mathrm{q}\right)$ is false, the truth value of p and q are respectively

• F, T

• F, F

• T, F

• T, T

Let p, q, r and s be statements and suppose that $\mathrm{p} \to \mathrm{q} \to \mathrm{r} \to \mathrm{p}$. If $~ \mathrm{s} \to \mathrm{r}$, then

• $\mathrm{s} \to ~ \mathrm{q}$

• $~ \mathrm{q} \to \mathrm{s}$

• $~ \mathrm{s} \to ~ \mathrm{q}$

• $\mathrm{q} \to ~ \mathrm{s}$