I need these please if you are not able to answer them all I want q3 please
math Jeff Edmonds York University Assignment 2 MATH1090 Predicate Logic I hope you find the course exhilarating and life changing. My answers all fit on these pages. Q1 Independent18 Q2 Tuples16 Q3 Matching7*8+10 Total100 1 Independent ie use only: " $ ⌐ Q1 Question 1: Question 2: Answer 1: Prove the statement by giving a strategy in which the prover always wins. eg For "c, you can say let c be an arbitrary value given to me by my adversary. Answer 2: Tuples Here the relation Loves(b,g) is True for 9 out of 16 of the tuples b,g. FTTF FTTF TFTF FTTT If I were to tell you that the following statement is true: $x5 $x1 "x4 $x2 "x6"x8 $x3 "x7 α(x1,x2,x3,x4,x5,x6,x7,x8) and tell you that each xi is chosen from the universe of n objects, then How many such tuples can be inputs to the relation α? What is the maximum number these tuples can α be True for? What is the minimum number these tuples can α be True for? What is the maximum number these tuples can α be False for? What is the minimum number these tuples can α be False for? For which of these answer is the fraction more than a half? Do we know if α is True for 0,0,0,0,0,0,0,0 or for 4,2,7,3,3,6,7,1? Explain all your answers. Q2 Tuples $x5 $x1 "x4 $x2 "x6"x8 $x3 "x7 α(x1,x2,x3,x4,x5,x6,x7,x8) Answer: Q2 Matching Game Q3 Matching Game Q3 Game (Race): Randomly choose two cards. Find an icon that appears on both. Matching Game Q3 Game (Race): Randomly choose two cards. Find an icon that appears on both. Matching Game Q3 Game (Race): Randomly choose two cards. Find an icon that appears on both. Matching Game Q3 Game (Race): Randomly choose two cards. Find an icon that appears on both. Our Universe is a set of these cards. Let appears(i,c) state that icon i appears on card c. Let $5 card c property(c) state that there are exactly five cards in our universe with the stated property. Matching Game Q3 Game (Race): Randomly choose two cards. Find an icon that appears on both. For this game to be fun to play, there is a necessarily property about the deck of cards as a whole that ensures that the pair of cards c1 & c2 chosen will in fact contain the icon i being searched for. State this property both in English and as a logic statement. ie use only: " $ ⌐ Question 1: Answer 1: Matching Game Q3 Consider the following universe consisting of 72 points p = x,y and all possible lines y=mx+b that go through them. Question 2: Answer 2: 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 State a property similar to that in Question 1, that holds about the universe of lines and points that ensures something useful about pairs of point p1 & p2 and a found line L. State this property both in English and as a logic statement. ie use only: " $ ⌐ Matching Game Q3 Question 3: Answer 3: 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Prove the statement from Question 2, by giving a strategy in which the prover always wins. Hint: Use (x2-x1)(y-y1)= (y2-y1)(x-x1) in your proof. Hint: Use “Proof by picture” in your proof. Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 From the figure, we see that lines with slope m=¼ are a little odd because they do not go through as many points as we would like. We will now move away from the model in which x and y are integers in {0,1,…,q-1} and instead consider a model in which x and y are integers {0,1,…,q-1} mod q. Here q is prime. "x&y{0,1,…,q-1}, x+y and xy are well defined and if x≠0 then m=1/x is also well defined. ie "x{1,…,q-1}, $1 m{1,…,q-1}, mx=1 For example, when q=7 we have 3+5=8=8-7=1 and 42=8=8-7=1. Rearranging the last equation gives that ¼=2. In fact, all of algebra in this world works as it does with the reals! 13 Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Color green all points in the line y=2x+4 (mod7) Explain at least one example. Question 4.1: Answer 4.1: Color green all points in the line y=¼x+4 (mod7) Explain at least one example. Question 4.2: Answer 4.2: Can the same two points have two different lines going through them Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Each line L is specified by an equation y=mx+b The possible values of the y-intercept b is: b{0,1,…,6} The possible values of the slope m is: m{0,1,…,6, }. The number of lines in our universe is: 78=56 icon/lines. Given any point p, there are 8 lines L passing through it. These correspond to the 8 possible slopes m m{0,1,…,6,∞}. Free Answers: ∞ y=mx+b x=2 Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Free Answers: y=mx+b x=2 One of my religions is to balance units. eg 5 (hrs) 100 (km/hr) = 500 (km) Form a bipartite graph. A node on the left for each of the 77 points. A node on the right for each of the 87 lines. An edge point,line if the point is on the line 778 point,line edges = 77 (points) = 87 (lines) … … 77 87 8 7 8 (lines/point) 7 (point/line) Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Question 5: Answer 5: State a property similar to that in Question 1 & 2, that holds about our universe of lines and points that ensures something useful about a line L, and integer value x{0,1,…,q-1}, and a found integer value y{0,1,…,q-1}. State this property both in English and as a logic statement. ie use only: " $ ⌐ m ∞ Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Question 6: Show how your answer to Question 4 is in fact a proof of your answer to Question 5. Answer 6: Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Question 7: Answer 7: Your answer to Question 5 proves another fun property that holds about our universe that ensures something useful about lines L and the number of point pj found in them. State this property both in English and as a logic statement. ie use only: " $ ⌐ Including m=∞ 19 Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 One point of formal proofs is to prove theorems with as few assumptions as possible about the nature of the objects we are talking about so that we can find a wide range of strange new objects for which the same theorems are true. Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Repeat your logic statement from Questions 1 & 2. Hopefully, you see the parallel between them. Use this connection to design a set of cards with cute pictures on each with which you can play this card game. How would you do this? Complete these numbers Question 8: ? icon,card edges = ? (cards) ? (icons/card) = ? (icons) ? (cards/icon) Matching Game Q3 0 1 2 3 4 5 6 q=7 0 1 2 3 4 5 6 Answer 8: