GATE-CS-2000

Question 21

Given the relations

employee (name, salary, deptno) and
department (deptno, deptname, address)

Which of the following queries cannot be expressed using the basic relational algebra operations (U, -, x, [Tex]\\pi[/Tex], [Tex]\\sigma[/Tex] , p)?

• Department address of every employee
• Employees whose name is the same as their department name
• The sum of all employees’ salaries
• All employees of a given department

Question 22

X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true?

• k must be 1
• k must be 0
• k can take any value between 0 and 1 (d) None of the above
• None of the above

Question 23

E1 and E2 are events in a probability space satisfying the following constraints:

 
    Pr(E1) = Pr(E2)
    Pr(EI U E2) = 1
    E1 and E2 are independent 

The value of Pr(E1), the probability of the event E1 is

• 0
• 1/4
• 1/2
• 1

Question 24

Which of the following statements is true?

• S > T
• S = T
• S < T and 2S > T
• 2S <= T

Question 25

A polynomial p(x) satisfies the following:

p(1) = p(3) = p(5) = 1 
p(2) = p(4) = -1 

The minimum degree of such a polynomial is

• 1
• 2
• 3
• 4

Question 26

A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following state­ments is true?

• R is not an equivalence relation
• R is an equivalence relation having 1 equivalence class
• R is an equivalence relation having 2 equivalence classes
• R is an equivalence relation having 3 equivalence classes

Question 27

Let P(S) denotes the power set of set S. Which of the following is always true?

• a
• b
• c
• d
• None

Question 28

Let a, b, c, d be propositions. Assume that the equivalences a ↔ (b V-b) and b ↔ c hold. Then the truth value of the formula (a ∧ b) → (a ∧ c) ∨ d) is always

• True
• False
• Same as the truth value of b
• Same as the truth value of d

Question 29

What can be said about a regular language L over {a} whose minimal finite state automaton has two states?

• L must be {aⁿ| n is odd}
• L must be {aⁿ| n is even}
• L must be {aⁿ| ³ O}
• Either L must be {aⁿ | n is odd}, or L must be {aⁿ | n is even}

Question 30

Consider the following decision problems:

(P1) Does a given finite state machine accept a given string
(P2) Does a given context free grammar generate an infinite 
     number of strings

Which of the following statements is true?

• Both (P1) and (P2) are decidable

• Neither (P1) nor (P2) are decidable

• Only (P1) is decidable

• Only (P2) is decidable