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14. Predicate and Quantifier                                                                        5 marks
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16. Q.1 The game of Draughts is played on a standard Chessboard 64 black and white chequered squares. Each player has 12 pieces (men) normally in the form of fat round counters. One player has black men and the other has white men. When starting, each player’s men are placed on the12 black squares nearest to that player (seeFigure). The white squares are not used at all in the game
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18. Figure: Starting position on a 8x8 Draughts board
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20. The men only move diagonally and so stay on the black squares throughout. Black always plays first. Players take turns to move a man of their own colour. There are fundamentally 4 types of move: the ordinary move of a man, the ordinary move of a king, the capturing move of a man and the capturing move of a king. An ordinary move of a man is its transfer diagonally forward left or right from one square to an immediately neighboring vacant square. When a man reaches the farthest row forward (the king-row or crown head) it becomes a king: another piece of the same shade is placed on top of the piece in order to distinguish it from an ordinary man. An ordinary move of a king is from one square diagonally forward or backward, left or right, to an immediately neighboring vacant square. Whenever a piece (man or king) has an opponent’s piece adjacent to it and the square immediately beyond the opponent’s piece is vacant, the opponent’s piece can be captured. If the player has the opportunity to capture one or more of the opponent’s pieces, then the player must do so. A piece is taken by simply hoppingoveritintothevacantsquarebeyondandremovingitfromtheboard. Unlike an ordinary move, a capturing move can consist of several such hops - if a piece takes an opponent’s piece and the new position allows it to take another piece, then it must do so straight away. Kings are allowed to move and capture diagonally forwards and backwards and are consequently more powerful and valuable than ordinary men. However, ordinary men can capture Kings. The game is won by the player who first manages to take all his opponent’s pieces or renders them unable to move.
21. For each of the following conditions on Draughts game write the corresponding axioms, using an appropriate first order logic language.
22. 1. Each piece is either white or black.
23. 2. Each piece is either a king or a man.
24. 3. White squares are always empty (always: in each instant of the game).
25. 4. In each instant of the game, black squares are either empty or contain a piece.
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27. To describe the above axioms, use given predicate
28. • square(x): "x is a square"
29. • piece(x): "x is a piece"
30. • white(x): "x is white"
31.  • black(x): "x is black"
32.  • man(x): "x is a man"
33. • king(x): "x is a king"
34. • empty(x,t): "square x is empty at time t"
35. • contain(x,y,t): "square x contains piece y at time t"
36.  • capture(x,y,t): "piece x captures piece y at time t"
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38. Rule of inference                                                                               5 marks
39. Q.2 Are the given argument logically correct?
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41. “If Carlo is fighting for his rights, he must be supported;
42.   Carlo is fighting for his rights. Thus he must be supported.”
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44. “If Carlo is fighting for his rights, he must be supported;
45. If Carlo is not fighting for his rights, he must not be supported;
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47.  “If Carlo is fighting for his rights, he must be supported;
48.  Carlo must not be Supported. Thus he is not fighting for his rights.”
49. iv)
50. “If Carlo is fighting for his rights, he must be supported;
51. He must be supported. Thus he is right.”
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55. SETS                                                                                            5 marks
56. Q.3 Fuzzy sets are used in artificial intelligence. Each element in the universal set U has a degree of membership, which is a real number between 0 and 1 (including 0 and 1), in a fuzzy set S. The fuzzy set S is denoted by listing the elements Links with their degrees of membership (elements with 0 degree of membership are not listed).
57. For instance, we write{0.6 Alice, 0.9 Brian, 0.4 Fred, 0.1 Oscar, 0.5 Rita} for the set F (of famous people) to indicate that Alice has a 0.6 degree of membership inF, Brian has a 0.9 degree of membership in F, Fred has a 0.4 degree of membership in F, Oscar has a 0.1 degree of membership in F, and Rita has a 0.5 degree of membership in F (so that Brian is the most famous and Oscar is the least famous of these people). Also suppose that R is the set of rich people with R ={0.4 Alice, 0.8 Brian, 0.2 Fred, 0.9 Oscar, 0.7 Rita}.
58. i) The complement of a fuzzy set S is the set S, with the degree of the membership of an element in S equal to 1 minus the degree of membership of this element in S. Find F (the fuzzy set of people who are not famous) and R (the fuzzy set of people who are not rich).
59. ii) The union of two fuzzy sets S and T is the fuzzy set S∪T, where the degree of membership of an element in S∪T is the maximum of the degrees of membership of this element in S and in T. Find the fuzzy set F∪R of rich or famous people.
60. iii) The intersection of two fuzzy sets S and T is the fuzzy set S∩T, where the degree of membership of an element in S∩T is the minimum of the degrees of membership of this element in S and in T. Find the fuzzy set F∩R of rich and famous people.
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62. iv) The Cartesian Product of two Fuzzy sets A and B is defined as
63.         AB(x,y) = min(A(x),B(x))
64. Let  
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67. then AB
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69. By given above example, Find (a) P = RI   (b) S = I N  
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73. Method of proof                                                                                 2 marks
74. Q.4 There is no positive integer solution of x2 - y2=1 for positive integer x and y
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76. Function                                                                                        3 marks
77. Q.5
78. i)      Classify each relation as a function, a one to one function or neither.
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83. ii)     Determine the given function is invertible or not.
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