Question 1 |

Consider the statement

"Not all that glitters is gold"

Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement?

"Not all that glitters is gold"

Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement?

\forall x:glitters(x)\Rightarrow \neg gold(x) | |

\forall x:gold(x)\Rightarrow glitters(x) | |

\exists x:gold(x)\wedge \neg glitters(x) | |

\exists x:glitters(x)\wedge \neg gold(x) |

Question 1 Explanation:

Question 2 |

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is

0.25 | |

0.5 | |

0.75 | |

1 |

Question 2 Explanation:

Question 3 |

Let G=(V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G ?

G_{1}=(V,E_{1}) where E_{1}\equiv \{(u,v)|(u,v)\notin E\} | |

G_{2}=(V,E_{2}) where E_{2}\equiv \{(u,v)|(v,u)\in E\} | |

G_{3}=(V,E_{3}) where E_{3}\equiv \{(u,v)| there is a path of lenth \leq 2 from u to v in E} | |

G_{4}=(V_{4},E) where V_{4} is the set of vertices in G which are not isolated |

Question 3 Explanation:

Question 4 |

Consider the following system of equations:

3x + 2y = 1

4x + 7z = 1

x + y + z = 3

x - 2y + 7z = 0

The number of solutions for this system is

3x + 2y = 1

4x + 7z = 1

x + y + z = 3

x - 2y + 7z = 0

The number of solutions for this system is

0 | |

1 | |

2 | |

3 |

Question 4 Explanation:

Question 5 |

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is

0 | |

1 | |

2 | |

3 |

Question 5 Explanation:

There are 5 questions to complete.