Question 1 |

Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A-B) \cup (B-A) \cup (A \cap B) is equal to

A \cup B | |

A^c \cup B^c | |

A \cap B | |

A^c \cap B^c |

Question 1 Explanation:

Question 2 |

Let X = \{2, 3, 6, 12, 24\}, Let \leq be the partial order defined by X \leq Y if x divides y. Number of edges in the Hasse diagram of (X, \leq) is

3 | |

4 | |

9 | |

None of the above |

Question 2 Explanation:

Question 3 |

Suppose X and Y are sets and |X| and |Y| are their respective cardinality. It is given that there are exactly 97 functions from X to Y. From this one can conclude that

|X| =1, |Y| =97 | |

|X| =97, |Y| =1 | |

|X| =97, |Y| =97 | |

None of the above |

Question 3 Explanation:

Question 4 |

Which of the following statements is FALSE?

The set of rational numbers is an abelian group under addition | |

The set of integers in an abelian group under addition | |

The set of rational numbers form an abelian group under multiplication | |

The set of real numbers excluding zero is an abelian group under multiplication |

Question 4 Explanation:

Question 5 |

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

\frac{1}{36} | |

\frac{1}{3} | |

\frac{25}{36} | |

\frac{11}{36} |

Question 5 Explanation:

Question 6 |

The formula used to compute an approximation for the second derivative of a function f at a point x_0 is

\dfrac{f(x_0 +h) + f(x_0 - h)}{2} | |

\dfrac{f(x_0 +h) - f(x_0 - h)}{2h} | |

\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 - h)}{h^2} | |

\dfrac{f(x_0 +h) - 2f(x_0) + f(x_0 - h)}{h^2} |

Question 6 Explanation:

Question 7 |

Let AX = b be a system of linear equations where A is an m \times n matrix and b is a m \times 1 column vector and X is an n \times 1 column vector of unknowns. Which of the following is false?

The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank. | |

If m < n and b is the zero vector, then the system has infinitely many solutions. | |

If m=n and b is a non-zero vector, then the system has a unique solution. | |

The system will have only a trivial solution when m=n, b is the zero vector and rank (A) =n. |

Question 7 Explanation:

Question 8 |

Which two of the following four regular expressions are equivalent? (\varepsilon is the empty string).

(i). (00)^ * (\varepsilon +0)

(ii). (00)^*

(iii). 0^*

(iv). 0(00)^*

(i). (00)^ * (\varepsilon +0)

(ii). (00)^*

(iii). 0^*

(iv). 0(00)^*

(i) and (ii) | |

(ii) and (iii) | |

(i) and (iii) | |

(iii) and (iv) |

Question 8 Explanation:

Question 9 |

Which of the following statements is false?

The Halting Problem of Turing machines is undecidable | |

Determining whether a context-free grammar is ambiguous is undecidable | |

Given two arbitrary context-free grammars G_1 and G_2it is undecidable whether L(G_1) = L(G_2) | |

Given two regular grammars G_1 and G_2 it is undecidable whether L(G_1) = L(G_2) |

Question 9 Explanation:

Question 10 |

Let L \subseteq \Sigma^* where \Sigma = \left\{a,b \right\}. Which of the following is true?

L = \left\{x \mid x \text{ has an equal number of } a\text{'s and }b\text{'s}\right \} is regular | |

L = \left\{a^nb^n \mid n \geq 1\right \} is regular | |

L = \left\{x \mid x \text{ has more number of }a\text{'s than }b\text{'s}\right \} is regular | |

L = \left\{a^mb^n \mid m \geq 1, n \geq 1 \right \} is regular |

Question 10 Explanation:

There are 10 questions to complete.