{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 0 "" }{TEXT 260 37 "Planes, T riangles, and Symbolic Logic" }}{PARA 19 "" 0 "" {TEXT -1 13 "J. A. Zi egler" }}{PARA 256 "" 0 "" {TEXT -1 55 "Web Page: http://www2.SPSU.ed u/math/ziegler/index.html" }}{PARA 256 "" 0 "" {TEXT -1 26 "E-mail: j ziegler@spsu.edu" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 17 "February 16, 2006" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 " Planes: The Interplay of Analysis and Geometry" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 603 "Becoming convinced that, say, 3x + 2y =4 is an equation of a line was very useful when one began to study systems of two such equations. The geometric possibilities were then obvious a nd they lead us to feel that our study of such systems was incomplete \+ until we had learned how to distinguish the possible cases analyticall y. As beginners, we were naturally disappointed and annoyed when, onl y a short time afterwards, we were simply told that, say, x + y + z = \+ 1 was the equation of a plane, but no visual evidence of this was prod uced. In those days, it simply was not possible, but not any more." } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 " What Sort of Graph Does a Lin ear Equation in Three Variables Have? " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Using Maple, we can produce th at evidence quite easily." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plots):with(plottools):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "First, we define an equation, then solve it for z. " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eq1:=x+y+z=1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Solving for z," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z1:=solve(eq1,z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Now we plot as points a number of solutions of this equa tion and display them together. So that the surface containing the so lutions is easier to see, the " }{TEXT 256 4 "grid" }{TEXT -1 178 " op tion has been used to increase the number of points used in our 20 x 2 0 plotting region from the default 25 x 25 to 100 x 100. This gives 1 0,000 points and very quickly, too." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "G:=plot3d(z1,x=-10..10,y=-10..10,grid=[100,100],colo r=green):\ndisplay(G,scaling=constrained,axes=normal,style=POINT);" }} }{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "Click on the picture. H olding down the left mouse button, move the mouse. Clearly, the graph is a plane. Using the mouse, we may change the style of the display \+ so that the graph may be seen more easily. On this evidence, we are n ow ready to consider visually the cases which arise from a 3 by 3 line ar system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 " A 3 x 3 System: The Possible Cases " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 " \+ A System with an Unique Solution" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Consider the sy stem" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=1;eq2:=x +y-z=1;eq3:=-2*x+3*y-z=4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Does this system have a solution?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Each equ ation represents a plane. The unique solution, ( " }{XPPEDIT 18 0 "- 1/5,6/5,0;" "6%,$*&\"\"\"F%\"\"&!\"\"F'*&\"\"'F%F&F'\"\"!" }{TEXT -1 190 " ), is the only point that lies on all three planes. To see thi s and the relationship between the planes, we graph them and use a sma ll sphere of a contrasting color to indicate the point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "z1:=solve(eq1,z):\nz2:=solve(eq2,z ):\nz3:=solve(eq3,z):\na:=plot3d(z1,x=-4..4,y=-2..4,color=yellow,scali ng=constrained):\nb:=plot3d(z2,x=-4..4,y=-2..4,color=red,scaling=const rained):\nc:=plot3d(z3,x=-4..4,y=-2..4,color=green,scaling=constrained ):\nS:=sphere([-1/5,6/5,0],.4,color=blue,style=patchnogrid):\ndisplay( a,b,c,S,view=[-4..4,-2..4,-5..5],style=patchcontour,scaling=constraine d,orientation=[75,60],axes=box);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 " Systems with No Solution" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 41 " Case 1: All of the Planes are Parallel" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "We consider the system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq1:=x+y+z=-6; eq2:=x+y+z=1; eq3:=x+y+z=8;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,e q3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "No output has been produced, which means that Maple cannot find a solution. " }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 68 "Again, we graph the planes defined by the equations of \+ our system. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "z1:=solve (eq1,z):\nz2:=solve(eq2,z):\nz3:=solve(eq3,z):\nA:=plot3d(z1,x=-10..10 ,y=-10..10,color=yellow,scaling=constrained):\nB:=plot3d(z2,x=-10..10, y=-10..10,color=red,scaling=constrained):\nC:=plot3d(z3,x=-10..10,y=-1 0..10,color=green,scaling=constrained):\ndisplay(A,B,C,style=patchcont our,view=[-10..10,-10..10,-10..15],scaling=constrained,orientation=[12 6,88],axes=box);" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "We se e why there is no solution. There is no point which lies on all three planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 41 " Case 2: \+ Two of the Planes are Parallel" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 " Analytical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=-3 ;eq2:=x+y+z=5;eq3:=x+y-5*z=-15;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Again, no output has been produced. Maple cannot find a s olution. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "To see why, we graph the planes de fined by the equations of our system. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "z1:=solve(eq1,z):\nz2:=solve(eq2,z):\nz3:=solve(eq3, z):\nA:=plot3d(z1,x=-10..10,y=-10..10,color=yellow,scaling=constrained ):\nB:=plot3d(z2,x=-10..10,y=-10..10,color=red,scaling=constrained):\n C:=plot3d(z3,x=-10..10,y=-10..10,color=green,scaling=constrained):\ndi splay(A,B,C,view=[-10..10,-10..10,-10..15],style=patchcontour,scaling= constrained,orientation=[128,80],axes=box);" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Again, there is no point which lies on all thre e planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 37 " Case 3: No Two Planes are Parallel" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "An alytical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:=x+y+z=-3;e q2:=x+y-z=1;eq3:=x+y-5*z=-25;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Again, Maple cannot find a solution. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We graph the planes defined by the equations of our system. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "z1:=solve(eq1,z):\nz2:=solv e(eq2,z):\nz3:=solve(eq3,z):\nA:=plot3d(z1,x=-10..10,y=-10..10,color=y ellow,scaling=constrained):\nB:=plot3d(z2,x=-10..10,y=-10..10,color=re d,scaling=constrained):\nC:=plot3d(z3,x=-10..10,y=-10..10,color=green, scaling=constrained):\ndisplay(A,B,C,view=[-10..10,-10..10,-10..15],st yle=patchcontour,scaling=constrained,orientation=[-56,87],axes=box);" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Once again, there is no point which lies on all three planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 " A System with an Infinite Number of Solutions " }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Analytical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq1:=x+y+z=1;eq2:=-x+z=0;eq3:=x+2*y+3*z=2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\}, \{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This shows that th ere is a solution for each value of x: hence, an infinite number of so lutions. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Graphical" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Following our usual prodedure, we \+ may see the geometrical relationships between the planes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "z1:=solve(eq1,z):\nz2:=solve(eq2,z ):\nz3:=solve(eq3,z):\nA:=plot3d(z1,x=-10..10,y=-10..10,color=yellow,s caling=constrained):\nB:=plot3d(z2,x=-10..10,y=-10..10,color=red,scali ng=constrained):\nC:=plot3d(z3,x=-10..10,y=-10..10,color=green,scaling =constrained):\ndisplay(A,B,C,view=[-10..10,-10..10,-10..15],style=pat chcontour,scaling=constrained,orientation=[121,85],axes=box);" }}} {EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "If we like, we can use our solutions to graph the line of intersection of the planes." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L:=spacecurve([z,-2*z+1,z],z =-5..6,color=blue,thickness=7):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display(A,B,C,L,view=[-10..10,-10..10,-10..15],style=patchcon tour,scaling=constrained,orientation=[121,85],axes=box);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 " Degenerate Cases" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1016 "In our discussion, we have assumed that the equations are of distinct planes. If this is not so, a few so-ca lled degenerate cases may arise. These are easiest to describe geomet rically. In the first, there is really only one plane and the equatio ns are then non-zero multiples of each other. If so, the coordinates \+ of every point in the plane is a solution. Second, there may be only \+ two distinct planes. It is geometrically obvious that these either int ersect along a line or are parallel. From the previous case, it is cl ear that two of the equations must be non-zero multiples of each othe r, but not of the third. Now suppose that this third equation is ax + by + cz = d. Only if there is a non-zero number, k, so that one of t he other equations can be written as kax + kby + kcz = d1 (the value o f d1 is of no interest), are the planes parallel. In this case, there are obviously no solutions. Otherwise, the planes intersect along a \+ line and the coordinates of each point on this line are solutions. " } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " Triangles: Seein g Them Helps" }}{EXCHG {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "To be able to solve a triangle, i t helps to be able to see it. The four functions given below form a c omplete Maple tool kit for exploring triangles. They allow us to see \+ the picture first, if there is one. At last! " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " Law of Sines Triangles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 48 " Two Sides and an Angl e adjacent to one of Them" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The f unction, T, defined below as a Maple " }{TEXT 257 9 "procedure" } {TEXT -1 514 ", is designed to take the usual \"two sides and an adjac ent angle\" information, in the usual order, \"angle-(adjacent-side)-( opposite-side),\" and draw a picture showing either the possible trian gle, or triangles, or that no triangle can be formed. If there is a t riangle, then the input angle must be, of course, an interior angle of the triangle. Appropriately, the function accepts as input an angle i n degrees. Maple's trigonometric functions accept only radian input, s o T converts the input angle to radians. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1394 "T:=proc(angle::positive,b::positive,a::positive)\nlocal A,B1,B2, S1,S2,P,Lc1,Lc2,Lb,Lc,L,Cir,pic,S,k,PT;\n A:=angle*Pi/180:\n \+ B1:=arcsin(b*sin(A)/a):\n B2:=Pi-B1:\n P:=b*cos(A),b*sin(A) ;\n Lb:=plot([t*cos(A),t*sin(A),t=0.001..b],color=blue,thickness= 3):\n Cir:=plottools[circle]([P],a,color=red,thickness=2):\n \+ PT:=plottools[disk]([P],a/20,color=red):\nif is(Pi/2-A,positive) an d is(b-a,positive) and is(a-b*sin(A),nonnegative) \nthen \n Lc1:= plottools[line]([0,0],[P[1]-a*cos(B2),P[2]-a*sin(B2)],color=blue,thick ness=3):\n Lc2:=plottools[line]([0,0],[P[1]-a*cos(B1),P[2]-a*sin( B1)],color=blue,thickness=3):\n Lc:=[Lc1,Lc2]:\n S1:=plot([P [1]-t*cos(B2),P[2]-t*sin(B2),t=0.001..a],color=green,thickness=4):\n \+ S2:=plot([P[1]-t*cos(B1),P[2]-t*sin(B1),t=0.001..a],color=green,th ickness=4):\n S:=[S1,S2]:\n pic:=seq(plots[display](S[k],Lc[ k],Lb,Cir,PT),k=1..2):\n plots[display](pic,insequence=true,scali ng=constrained);\nelif is(Pi-A,positive) and is(a-b,positive) \nthen \n Lc1:=plottools[line]([0,0],[P[1]-a*cos(B2),P[2]-a*sin(B2)],co lor=blue,thickness=3):\n S1:=plot([P[1]-t*cos(B2),P[2]-t*sin(B2), t=0.001..a],color=green,thickness=4):\n plots[display](S1,Lc1,Lb, Cir,PT,scaling=constrained);\nelse \n L:=plottools[line]([0,0],[ b/4,0],color=blue,thickness=3):\n plots[display](Lb,Cir,PT,L,scal ing=constrained);\nfi;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "F irst, an example of the \"acute angle, one triangle\" case." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "T(45,5,7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Second, an example of \"the ambiguous case.\" This is animated. The circle shows clearly that two triangles are p ossible." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "T(30,5,4);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Next, an example of the \"obtuse a ngle\" case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T(120,5,7); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Finally, two examples of \"n o triangle\" cases, first, with an acute angle and, second, with an ob tuse one. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T(60,15,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T(135,5,4);" }}}{EXCHG } {EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Notice particularly in this last \+ example the importance of remembering that the angle must be an interi or angle of the proposed triangle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 27 " The Angle-Side- Angle Case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "The function ASA ac cepts input in the form \"angle-side-angle,\" where the angles are in \+ degrees, and either draws the triangle determined by this data or repo rts that it does not determine a triangle." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 495 "ASA:=proc(angle1::positive,c::positive,angle2::pos itive)\nlocal A,B,C,a,b,P,La,Lb,Lc;\n A:=angle1*Pi/180:\n B: =angle2*Pi/180:\n C:=Pi-(A+B):\nif is(C,positive)\nthen \n \+ b:=c*sin(B)/sin(C):\n P:=b*cos(A),b*sin(A):\n Lc:=plottools[ line]([0,0],[c,0],color=blue,thickness=3):\n Lb:=plottools[line]( [0,0],[P],color=red,thickness=3):\n La:=plottools[line]([c,0],[P] ,color=green,thickness=3):\n plots[display](La,Lb,Lc,scaling=cons trained);\nelse `NO TRIANGLE`\nfi;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Two simple cases: Two acute angles " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ASA(30,5,45);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "One acute angle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ASA(120,3,40);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are no triangles with two obtuse angles, right?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ASA(120,7,100);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 " Law of Cosines Triangles " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 " The Side-Angle-Side Case " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "The function SAS accepts input in the form \"side-angle-side,\" where the angle is in degrees, and eith er draws the triangle determined by this data or reports that it does \+ not determine a triangle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "SAS:=proc(a::positive,angle::positive,b::positive)\nlocal C,c,La, Lb,Lc,P,S;\n C:=angle*Pi/180:\nif is(Pi-C,positive)\nthen\n \+ c:=sqrt(a^2+b^2-2*a*b*cos(C)):\n P:=b*cos(C),b*sin(C):\n La: =plottools[line]([0,0],[a,0],color=red,thickness=3):\n Lb:=plotto ols[line]([0,0],[P],color=blue,thickness=3):\n Lc:=plottools[line ]([P],[a,0],color=green,thickness=3):\n plots[display](La,Lb,Lc,s caling=constrained);\nelse `NO TRIANGLE`\nfi;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "If the included angle is 130 degrees" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "SAS(5,130,7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "If the included angle is acute" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "SAS(5,75,8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "If the included angle is too large" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "SAS(5,230,7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 " Three Sides" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Finally, the function SSS accepts the lengths \+ of three line segments and either draws the triangle determined by thi s data or reports that it does not determine a triangle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "SSS:=proc(a::positive,b::positive, c::positive)\nlocal C,La,Lb,Lc,P;\nif a+b>c and a+c>b and b+c>a then\n C:=arccos((a^2+b^2-c^2)/(2*a*b)):\n P:=b*cos(C),b*sin(C):\n La:=plottools[line]([0,0],[a,0],color=red,thickness=3):\n L b:=plottools[line]([0,0],[P],color=blue,thickness=3):\n Lc:=plott ools[line]([P],[a,0],color=green,thickness=3):\n plots[display](L a,Lb,Lc,scaling=constrained);\nelse `NO TRIANGLE`\nfi;\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Sometimes there is a triangle (if \+ the sum of any two sides is greater than the third)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "SSS(5,4,6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "And sometimes there is not." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "SSS(1,2,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 " Symbolic Logic: The Tr uth of It" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 18 "Typical Statements" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "In the following, p and q are \"logical \+ variables\", with the values \"true\" or \"false\". Statements compos ed from p and q, for example, \"q or (not p)\", we will call 2-stateme nts. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 " The following are typi cal 2-statements from symbolic logic. We may use Maple to find and di splay their \"truth values\" in a \"truth table\"." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "1. p and q" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "2. p or q" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "3. not (p and \+ q)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "4. not (p or q)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "5. (not p) or (not q)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "6. (not p) and (not q)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "7. q or (not p) [This is \"If p, then q.\" ]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "8. (q or (not p) ) and (p o r (not q) ) [This is \"p if, and only if, q.\"]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "9. not (not p and q) and (p or q)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "10. not ((not p or q) and p) or q" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 73 " Truth Tables: The Truth Value of a Statement of Two Lo gical Variables " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "To generate th e truth table for 2-statements, we define the function " }{TEXT 258 3 "TT2" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "T T2:=proc(s::boolean)\n local P,Q,S,B;\n S:=subs(p=P,q=Q,s):\n \+ print(`p q s`);\nfor P in [true,false] do\n for Q in \+ [true,false] do\n B:=evalb(S):\n print(P,Q,B);\n od;\nod; \nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 208 "To generate the truth \+ table for any of the 2-statements, above, copy and paste it as the arg ument of TT2. The truth values of the 2-statement appear under \"s\", for \"statement\". For example, using 1, we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "TT2( p or q );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0p~~~~~~q~~~~~~sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%% %trueGF#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%trueG%&falseGF#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%&falseG%%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&falseGF#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "If one likes, one may mak e up one's own statement in p and q and, using the others as a model, \+ type it in as the argument of TT2." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Truth tables for three or more variables can be produced by a s imple extension of this procedure." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 " Logical Equivalence" }}{PARA 0 "" 0 "" {TEXT 259 110 "Two st atements are said to be logically equivalent if they have the same \"t ruth value\" in every possible case." }{TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 " Comparing the Truth Values of Two 2-Statemen ts" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "To generate a truth table to compare two 2-statements, use the following function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "E2:=proc(s1::boolean,s2::boolean) \n local P,Q,S1,S2,B1,B2;\n S1:=subs(p=P,q=Q,s1):\n S2:=subs (p=P,q=Q,s2):\n print(`p q s1 s2`); \n for P in [ true,false] do\n for Q in [true,false] do\n B1:=evalb(S1): \n B2:=evalb(S2):\n print(P,Q,B1,B2)\n od;\nod;\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The negation of \"p and q\", is \" (not p) or (not q)\". We may use our function E2 to check on this." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "E2(not(p and q),(not p) or (not q));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 71 " Determ ining the Equivalence of Statements of Three Logical Variables" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Each of the following statements \+ has three \"logical variables.\" Use the procedure below, as before, \+ to determine which of them are equivalent." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "1. p and (q or r)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "2. (p and q) or (p and r)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "3. p or (q and r)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "4. (p or \+ q) and (p or r)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 347 "E3:=pro c(s1::boolean,s2::boolean)\n local P,Q,R,S1,S2,B1,B2;\n S1:=subs (p=P,q=Q,r=R,s1):\n S2:=subs(p=P,q=Q,r=R,s2):\n print(` p q r s1 s1`);\n for P in [true,false] do\n for Q in [tr ue,false] do\n for R in [true,false] do\n B1:=evalb(S1):\n \+ B2:=evalb(S2):\n print(P,Q,R,B1,B2)\n od;\n od;\nod;\ne nd: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E3(p and (q or r),( p and q) or (p and r));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "The way is now clear to comp aring statements of four or more variables by continued simple modific ations of the E2 procedure. With Maple's help, the comprison will req uire hardly any effort at all." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Remark" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 551 "G. W. Leibniz was one of the founders of the calculus, but, among many other contributions of knowledge, he was also the fou nder of symbolic logic. Deeply impressed by the way in which algebrai c notation had \"mechanized\" mathematical arguments about numbers, he conceived the idea of using symbols to facilitate, in a similar way, \+ deductive arguments in logic. The details had to wait until the work o f George Boole and Agustus DeMorgan in the 19th century, but I like to think that he would have been delighted with what Maple has just help ed us do." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1 0 0" 48 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }