add calculations for false properties to AT5 (#947)

* add calculations for false properties to AT5

* remove tedious cube scenario

* remove unused falsehard variable

* use assertion rather than printing warning

* show example calculations for non-properties

* remove unused sample values

Author: StevenClontz

source/linear-algebra/exercises/outcomes/AT/AT5/generator.sage

@@ -8,39 +8,61 @@ class Generator(BaseGenerator):
         v2 = vector([x2,y2])
         v3 = vector([x3,y3])
         v = vector([x,y])
+        cs, ds = 2,3
+        v1s = vector([1,2])
+        v2s = vector([2,3])
+        v3s = vector([3,4])
+        vs = vector([1,2])
         vectorsimplify = lambda v : vector([simplify(expand(x)) for x in v])
 
         true_property_options = ["add_assoc","add_comm","mul_assoc","dist_v","dist_s","mul_id"]
         false_only_property_options = ["add_id","add_inv"]
 
-        def verify(plus,times,hardfalseproperties=[]):
+        def verify(plus,times):
             trueproperties={}
-            falseproperties=hardfalseproperties
+            falseproperties={}
             for prop in true_property_options:
                     if prop == "add_assoc":
                         LHS = plus(v1,plus(v2,v3))
                         RHS = plus(plus(v1,v2),v3)
+                        LHS_sample = plus(v1s,plus(v2s,v3s))
+                        RHS_sample = plus(plus(v1s,v2s),v3s)
                     elif prop == "add_comm":
                         LHS = plus(v1,v2)
                         RHS = plus(v2,v1)
+                        LHS_sample = plus(v1s,v2s)
+                        RHS_sample = plus(v2s,v1s)
                     elif prop == "mul_assoc":
                         LHS = times(c*d,v)
                         RHS = times(c,times(d,v))
+                        LHS_sample = times(cs*ds,vs)
+                        RHS_sample = times(cs,times(ds,vs))
                     elif prop == "mul_id":
                         LHS = times(1,v)
                         RHS = v
+                        LHS_sample = times(1,vs)
+                        RHS_sample = vs
                     elif prop == "dist_v":
                         LHS = times(c,plus(v1,v2))
                         RHS = plus(times(c,v1),times(c,v2))
+                        LHS_sample = times(cs,plus(v1s,v2s))
+                        RHS_sample = plus(times(cs,v1s),times(cs,v2s))
                     elif prop == "dist_s":
                         LHS = times(c+d,v)
                         RHS = plus(times(c,v),times(d,v))
+                        LHS_sample = times(cs+ds,vs)
+                        RHS_sample = plus(times(cs,vs),times(ds,vs))
                     LHS = vectorsimplify(LHS)
                     RHS = vectorsimplify(RHS)
                     if LHS == RHS:
-                        trueproperties[prop]=vectorsimplify(LHS)
+                        trueproperties[prop]=LHS
                     else:
-                        falseproperties.append(prop)
+                        falseproperties[prop]={
+                            "LHS": LHS, 
+                            "RHS": RHS, 
+                            "LHS_sample": LHS_sample,
+                            "RHS_sample": RHS_sample
+                        }
             for prop in false_only_property_options:
                 if "dist_s" in trueproperties and "mul_id" in trueproperties:
                     if prop == "add_id":
@@ -54,9 +76,6 @@ class Generator(BaseGenerator):
                             if vectorsimplify(LHS) != vectorsimplify(RHS):
                                 falseproperties.append("add_inv")
             return (trueproperties, falseproperties)
-        
-        #Use this to code in false properties that cannot be checked automatically ("add_id" and "add_inv")
-        hardfalseproperties=[]
 
         #Use this to list a property that is true, but you don't want students to check
         #because it is too easy (usually "add_comm") or too hard
@@ -73,8 +92,6 @@ class Generator(BaseGenerator):
             theta = lambda v : vector([v[0]+a,v[1]+b])
             untheta = lambda v : vector([v[0]-a,v[1]-b])
 
-            hardfalseproperties += ["add_id","add_inv"]
-
         elif n==1:
             plus = lambda v1,v2 : vector([v1[0]+v2[0], v1[1]+v2[1]])
             r1 = randrange(1,9)
@@ -87,9 +104,7 @@ class Generator(BaseGenerator):
 
         elif n==2:            
             plus = lambda v1,v2 : vector([v1[0]+v2[0], v1[1]+v2[1]])
-            r2 = randrange(2,4)
-            #times= lambda c,v : vector([c*v[0],c^(r2)*v[1]])
-            times= lambda c,v : vector([c^(r2)*v[0],c*v[1]])
+            times= lambda c,v : vector([c^2*v[0],c*v[1]])
             a=randrange(1,8)
             b=randrange(2,8)
             theta = lambda v: vector([v[0]+b*v[1],v[1]+a])
@@ -131,12 +146,12 @@ class Generator(BaseGenerator):
         oplus = lambda v1,v2 : theta(plus(untheta(v1),untheta(v2)))
         otimes = lambda c,v : theta(times(c,untheta(v)))
 
-        trueproperties, falseproperties = verify(oplus,otimes,hardfalseproperties)
+        trueproperties, falseproperties = verify(oplus,otimes)
+        
         for prop in true_no_check_properties:
-            if prop in trueproperties.keys():
-                trueproperties.pop(prop)
-            else:
-                print("WARNING:  Property "+prop + " was false.")
+            assert prop in trueproperties.keys()
+            trueproperties.pop(prop)
+        
         trueproperty, verification = choice(list(trueproperties.items()))
 
         return {
@@ -145,5 +160,5 @@ class Generator(BaseGenerator):
             "trueproperty": {
                 trueproperty: verification
             },
-            "falseproperties": {f: True for f in falseproperties},
+            "falseproperties": falseproperties,
         }

source/linear-algebra/exercises/outcomes/AT/AT5/template.xml

@@ -47,30 +47,6 @@ Show both sides simplify to <m>{{add_comm}}</m>.
         </outtro>
     </knowl>
     <!-- {{/add_comm}} -->
-    <!-- {{#add_id}} -->
-    <!-- <knowl>
-        <content>
-            <p>
-Show that there exists an additive identity element, that is:
-                <me>
-\text{There exists }(w,z)\in V
-\text{ such that }(x,y)\oplus(w,z)=(x,y).
-                </me>
-            </p>
-        </content>
-        <outtro><p>TODO</p></outtro>
-    </knowl> -->
-    <!-- {{/add_id}} -->
-    <!-- {{#add_inv}} -->
-    <!-- <knowl>
-        <content>
-            <p>
-Show that additive inverses exist.
-            </p>
-        </content>
-        <outtro><p>TODO</p></outtro>
-    </knowl> -->
-    <!-- {{/add_inv}} -->
     <!-- {{#mul_assoc}} -->
     <knowl>
         <content>
@@ -160,14 +136,20 @@ any one of the following properties does not hold:
                 <!-- {{#add_assoc}} -->
                 <item>
                     <p>
-Vector addition is not associative.
+Vector addition is not associative: <me>(x_1,y_1)\oplus((x_2,y_2)\oplus(x_3,y_3))={{LHS}}</me>
+<me>((x_1,y_1)\oplus(x_2,y_2))\oplus(x_3,y_3)={{RHS}}</me>
+For example: <me>(1,2)\oplus((2,3)\oplus(3,4))={{LHS_sample}}</me>
+<me>((1,2)\oplus(2,3))\oplus(3,4)={{RHS_sample}}</me>
                     </p>
                 </item>
                 <!-- {{/add_assoc}} -->
                 <!-- {{#add_comm}} -->
                 <item>
                     <p>
-Vector addition is not commutative.
+Vector addition is not commutative: <me>(x_1,y_1)\oplus(x_2,y_2)={{LHS}}</me>
+<me>(x_2,y_2)\oplus(x_1,y_1)={{RHS}}</me>
+For example: <me>(1,2)\oplus(2,3)={{LHS_sample}}</me>
+<me>(2,3)\oplus(1,2)={{RHS_sample}}</me>
                     </p>
                 </item>
                 <!-- {{/add_comm}} -->
@@ -188,28 +170,42 @@ Additive inverses do not always exist.
                 <!-- {{#mul_assoc}} -->
                 <item>
                     <p>
-Scalar multiplication is not associative.
+Scalar multiplication is not associative: <me>c\odot(d\odot(x,y))={{LHS}}</me>
+<me>(cd)\odot(x,y)={{RHS}}</me>
+For example: <me>2\odot(3\odot(1,2))={{LHS_sample}}</me>
+<me>(2\cdot 3)\odot(1,2)={{RHS_sample}}</me>
                     </p>
                 </item>
                 <!-- {{/mul_assoc}} -->
                 <!-- {{#mul_id}} -->
                 <item>
                     <p>
-<m>1</m> is not a scalar multiplication identity.
+<m>1</m> is not a scalar multiplication identity: <me>1\odot(x,y)={{LHS}} \neq (x,y)</me>
+For example: <me>1\odot(1,2)={{LHS_sample}} \neq (1,2)</me>
                     </p>
                 </item>
                 <!-- {{/mul_id}} -->
                 <!-- {{#dist_v}} -->
                 <item>
                     <p>
-Scalar multiplication does not distribute over vector addition.
+Scalar multiplication does not distribute over vector addition:
+<me>c\odot((x_1,y_1)\oplus(x_2,y_2))={{LHS}}</me>
+<me>(c\odot(x_1,y_1))\oplus(c\odot(x_2,y_2))={{RHS}}</me>
+For example:
+<me>2\odot((1,2)\oplus(2,3))={{LHS_sample}}</me>
+<me>(2\odot(1,2))\oplus(2\odot(2,3))={{RHS_sample}}</me>
                     </p>
                 </item>
                 <!-- {{/dist_v}} -->
                 <!-- {{#dist_s}} -->
                 <item>
                     <p>
-Scalar multiplication does not distribute over scalar addition.
+Scalar multiplication does not distribute over scalar addition:
+<me>(c+d)\odot(x,y)={{LHS}}</me>
+<me>(c\odot(x,y))\oplus(d\odot(x,y))={{RHS}}</me>
+For example:
+<me>(2+3)\odot(1,2)={{LHS_sample}}</me>
+<me>(2\odot(1,2))\oplus(3\odot(1,2))={{RHS_sample}}</me>
                     </p>
                 </item>
                 <!-- {{/dist_s}} -->