Consider the functions

I. $${e^{ - x}}$$

II. $${x^2} - \sin x$$

III. $$\sqrt {{x^3} + 1} $$

Which of the above functions is/are increasing everywhere in [0,1]?

Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x.

For n > 2, let a {0, 1}ⁿ be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1}ⁿ.
Then, the probability that $$\sum\limits_{i = 1}^n {{a_i}{x_i}} $$ is an odd number is _______.

Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.