GATE-CS-2003

Question 31

Let (5, ≤) be a partial order with two minimal elements a and b, and a maximum element c.

Let P : S → {True, False} be a predicate defined on S.
Suppose that P(a) = True, P(b) = False and 
P(x) ⇒ P(y) for all x, y ∈ S satisfying x ≤ y, 
where ⇒ stands for logical implication.

Which of the following statements CANNOT be true ?

• P(x) = True for all x ∈ S such that x ≠ b
• P(x) = False for all x ∈ S such that x ≠ a and x ≠ c
• P(x) = False for all x ∈ S such that b ≤ x and x ≠ c
• P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

Question 32

Which of the following is a valid first order formula ? (Here α and β are first order formulae with x as their only free variable)

• A
• B
• C
• D

Question 33

Consider the following formula a and its two interpretations I1 and I2 

Which of the following statements is true?

• I2 satisfies α, I1 does not
  

• Neither I2 nor I2 satisfies α
  

• Both I1 and I2 satisfy α
  

• I1 satisfies α, I2 does not
  

Question 34

m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥ 1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls? 

• C
  

• D
  

• B
  

• A
  

Question 35

Consider the following recurrence relation The value of T(m²) for m ≥ 1 is

• (m/6) (21m - 39) + 4
• (m/6) (4m² - 3m + 5)
• (m/2) (m^2.5 - 11m + 20) - 5
• (m/6) (5m³ - 34m² + 137m - 104) + (5/6)

Question 36

How many perfect matchings are there in a complete graph of 6 vertices ?

• 15
• 24
• 30
• 60

Question 37

Let f : A → B be an injective (one-to-one) function.

Define g : 2A → 2B as :
g(C) = {f(x) | x ∈ C}, for all subsets C of A.
Define h : 2B → 2A as :
h(D) = {x | x ∈ A, f(x) ∈ D}, for all subsets D of B. 

Which of the following statements is always true ?

• g(h(D)) ⊆ D
• g(h(D)) ⊇ D
• g(h(D)) ∩ D = ф
• g(h(D)) ∩ (B - D) ≠ ф

Question 38

Consider the set {a, b, c} with binary operators + and × defined as follows :

+	a	b	c		×	a	b	c
a	b	a	c		a	a	b	c
b	a	b	c		b	b	c	a
c	a	c	b		c	c	c	b

For example, a + c = c, c + a = a, c × b = c and b × c = a. Given the following set of equations :

(a × x) + (a × y) = c
(b × x) + (c × y) = c

The number of solution(s) (i.e., pair(s) (x, y)) that satisfy the equations is :

• 0
• 1
• 2
• 3

Question 39

Let ∑ = (a, b, c, d, e) be an alphabet. We define an encoding scheme as follows : g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11. Which of the following numbers is the encoding h of a non-empty sequence of strings ?

• 2⁷ 3⁷ 5⁷
• 2⁸ 3⁸ 5⁸
• 2⁹ 3⁹ 5⁹
• 2¹⁰ 5¹⁰ 7¹⁰

Question 40

A graph G = (V, E) satisfies |E| ≤ 3 |V| - 6. The min-degree of G is defined as . Therefore, min-degree of G cannot be

• 3
• 4
• 5
• 6