Which of the following proposition is a tautology?

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

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

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

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

If N denote the set of all natural numbers and R be the relation on N $\times$ N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d), then R is

• symmetric only

• reflexive only

• transitive only

• an equivalence relation

The function f : R $\to$ R defined by f (x) = (x - 1) (x - 2) (x - 3) is

• one-one but not onto

• onto but not one-one

• both one-one and onto

• neither one-one nor onto

The relation R defined on the set of natural numbers as { (a, b) : a differs from b by 3} is given

• {(1, 4), (2, 5), (3, 6), ... }

• {(4, 1), (5, 2), (6, 3), ... }

• {(1, 3), (2, 6), (3, 9), ... }

• None of the above

If R be a realtion from A = {1, 2, 3, 4} to B = {1, 3, 5} such that (a, b) ∈R $\iff \mathrm{a} < \mathrm{b}$, then ROR^-1 is

• {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

• {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}

• {(3, 3), (3, 5), (5, 3), (5, 5)}

• {(3, 3), (3, 4), (4, 5)}

Let f : R $\to$ R be defined as f(x) = x² + 1, find f^-1(- 5).

• $\left\{\mathrm{\varphi }\right\}$

• $\mathrm{\varphi }$

• $\left\{5\right\}$

• $\left\{- 5, 5\right\}$

If F is function such that F (0) = 2, F(1) = 3, F(x + 2)= 2F(x) - F(x + 1) for x > 0, then F(5) is equal to

• - 7

• - 3

• 17

• 13

Let S be a set containing n elements. Then, number of binary operations on S is

• nⁿ

• $2^{{\mathrm{n}}^2}$

• ${\mathrm{n}}^{{\mathrm{n}}^2}$

• n²

If f : [2, 3] $\to$ R is defined by f(x) = x³ + 3x - 2, then the range of f(x) is contained in the interval

• [1, 12]

• [12, 34]

• [35, 50]

• [- 12, 12]

If R be a relation defined as aRb iff $\left|\mathrm{a} - \mathrm{b}\right|$ > 0, then the relation is

• reflexive

• symmetric

• transitive

• symmetric and transitive