JN=00380327

please in the assignment you will need a j number above thank you.

j1 =0 j2=0 J3=3 J4=8

j5=0 J6=3 j7=2 J8=7

6:56 270Exam1A V Done 8 5 of 30 103 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A R Note that f, g and h are real numbers that are not all zero and that StLn is a one-place predicate that decides if the set S is a straight line or not. (1) Given two distinct points P = (P1, P2) E R2 and Q := (q1, 92) ER ( k + 1 # 0) SL P . Q = ( x , y ) ER2 3k, ER A x = P1 + 191 ) (y = kp2 + 192 R Note that (Pittqi , "P2+19z ) are the co-ordinates of the point that divides the straight-line segment joining P and Q in the ratio k: l. The division can be internal or external depending on the actual values of k and I. R Note that SL P, Q is the name of a set. SL P, Q is one possible definition of the set of points that constitute the straight line in the plane passing through the points P and Q. (2) Given any subset S of the plane R2 and (P, Q) E S2 \ As (that is two distinct points P and Q in S) we define S to be a straight line in the second sense as follows: StLn2 S :|VP, Q E S2 \As S = SL ( P, Q) R The purpose of problems (1c), (2c), and (3c) is to prove that the two definitions of straightlinehood agree. R Do all work on pages following the page on which the problem appears and number them as indicated. You may use the blank 104 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A man that Q A ...6:56 270Exam1A V Done 101 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A R The numbers a, b, c and d, as defined below, will occur throughout the exam, so please make sure that they are correct. Otherwise, all your answers will be wrong. D Your J-number J:= := (x|K E 1..8) Hence 12 = J3 = JA= Js = 16 = 17 = J8 = D mnm := minimum mxm := maximum D a := mnm JK E O. . 9 JK # 0 D b := mxm Jk k E 1.8 +1 = ( )+1 = D c := mnm ke 1.. 8 Jk * of D d := mxm ke 1..8 Jk = 0 R Please talk to me if a, b, and c are not distinct numbers. 102 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A (-1 if Jk(mod(3)) = 1 Q A ...6:56 I '9' @' 270Exam1A v Done You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A 1 if ]k(mad(3))=1 0 if ]k(mad(3))=0 1 if ]k(mod(3))=2 D Define VkE 1..8 Ak Show calculations on the facing side. Hence A1 = A2 = A3 = A4 = A5 A6: A7: A8: D Definitions for (1c), (2c), (3c): (0) Given any subset S of the plane R2 we define S to be a straight line in the first sense if and only if (fx+gy+h=0) A 5 == (96.)!) E R2 716R} 0091,05 R3 \[(0,0,71)E 1K3 We write: VS 6 TUE?) 5an1 (s) :4: (30,51,106 [R3 \{(0,0,n) e R3 n E MKS :: ,(XJ) G R: (f): + gy + h 2 MD). R In words, a S is a straight line in the first sense, if and only if, S is the solution set in R2 of some equation fx + gy + h = 0 where at least one of the coefficients is not 0. The condition rules out the possibility that the equation is 0 = 0 or 0 = n where n is a non- zero real number. 103 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A R Note that f, g and h are real numbers that are not all zero and that Sth1 is a one-place predicate that decides if the set S is a straight line or not. {'H Ciunn turn riinh'nrt nnintc D := {1L 11_\ t: m2 and n :: (nl n_\ C D2 Q o [a 6-57 I '0' E) 270Exam1A v Done 105 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 1A 6 (1a) Compute the following, recording only the answers on this page and showing all computations on pages following this one numbered 1070, 1071 etc. using as many as you need: a d 0 1(1) A :=|0 b 0' 006 2 101) A2 = [ 1 = 3 1(iii) A3 = [ 1 = 3 (iv) A3 (a + b + c)A2 + (ab + be + ca)A (abc)l3 No points on this page if answer to part (iv) (which acts as a check) for the work is incorrect. 106 You may talk to anyone Due by 2359 hours on 20230919 270 Exam 'lA Q o a