Download College Geometry University of Memphis MATH 3581 Mathematical

College Geometry
MATH 3581
Spring 2011
University of Memphis
Mathematical Sciences
Dwiggins
Homework Assignment # 1
Logical Systems
Page 3, # 7. A = A duck is a bird, or If x is a duck then x is a bird, which we can take as a true
statement. The converse of A would be If x is a bird then x is a duck, or Every bird is a duck,
which is a false statement. Thus, in the words of the textbook, A is not reversible.
B = A line is on a plane if and only if at least two points of the line are on the plane.
This is a biconditional statement, which is equal to its own converse, as are E, F, G, H, which
makes the instructions unclear. But we can look at both implications and see if either are true.
The “only if” part: If ℓ is a line on a plane then at least two points of the line are on the plane.
This is true if we assume in the first part that “on a plane” means every point on ℓ is in the plane.
The “if” part: If at least two points on ℓ are in a plane then ℓ is completely in the plane.
This is true if we assume it as one of the Euclidean postulates.
Thus, since the biconditional B is true in either direction, B is a true statement.
C = If a circle is on a plane (i.e. each point on the circle is in the plane), then at least two points
of the circle on the plane. Obviously, C is true. The converse is not. You can bisect a circle
with a plane  which is perpendicular to the plane of the circle, and there will be two points of
intersection, but the circle does not lie completely in .
D = The boy is a McCoy if he has red hair, i.e. If a boy has red hair then the boy is a McCoy.
The converse of D is If a boy is a McCoy then the boy has red hair, i.e. The boy has red hair if he
is a McCoy. I sincerely doubt either statement is true.
E = The equality of the lengths of two sides of a quadrilateral is necessary and sufficient for the
quadrilateral to be a square. You can see this to be false by considering one vertex of the
quadrilateral being at the origin of an xy coordinate system. Let two sides of the quadrilateral be
of unit length along the positive x and y axes, so you have three vertices with coordinates (0,0),
(1,0), and (0,1). Now let the fourth point be any point (a,b) in the first quadrant, and you can see
the quadrilateral can be made in any shape you want.
Thus, having two sides the same length in a quadrilateral is not sufficient for it to be a square. It
is a necessary condition, however. That is, the statement, If a quadrilateral is a square then it has
two sides of the same length, is true because in a square all four sides have the same length.
F = Rat is tar if and only if tar is rat. This is true by the associativity of multiplication. 
G = A circle is a square if and only if two radii are equal. This is false because the first part is
always false and the second part is always true, and a biconditional statement is true only when
both parts have the same value.
H = A necessary and sufficient condition for a rectangle to be a square is that the diagonals are
equal. This is also a biconditional statement.
Necessity: If a rectangle is a square then its diagonals are equal. This is true.
Sufficiency: If a rectangle has its diagonals equal then the rectangle is a square. This is false
because the first part is always true but the second part doesn’t have to be true.
Thus H is a false statement.
Page 6, # 5.
A = If two lines are parallel then they do not intersect.
This is true, as is the contrapositive: If two lines intersect then they are not parallel.
The converse of A is If two lines do not intersect then they are parallel, which is false unless it is
also assumed the two given lines are coplanar, which they do not have to be in 3D geometry.
The inverse of A, If lines are not parallel then they must intersect, is also false in 3D geometry.
B = If angles A and B of a triangle ABC are equal, the triangle is isosceles.
This is true because sides opposite equal angles have equal length, and what is also true is that
angles opposite equal sides in a triangle must have the same measure. However, this does not
mean the converse of B is true, because it is too specific about which two angles are equal.
The converse of B is If triangle ABC is isosceles then mA = mB, which does not have to be
true. We might instead have mA = mC. If this is true then the inverse of B, If mA  mB
then triangle ABC can’t be isosceles, is also false.
The contrapositive of B is true, however: If a triangle is not isosceles (i.e. if it is scalene) then no
two of its angles can have the same measure.
C = If a triangle is on a Euclidean plane (i.e if ABC lies in plane ), its angle sum is 180º.
This is true as a result of the Euclidean version of the Parallel Postulate. Its converse is also true,
on a deeper level than you might at first think. If a triangle has an angle sum equal to 180º, i.e.
equal in measure to a straight angle, then the plane it uniquely defines must be the Euclidean
plane. If instead the triangle were on the surface of a sphere then its angle sum would be greater
than 180º, and if we were in hyperbolic geometry the angle sum would be less than 180º. The
comparison to the angle sum in a triangle to that of a straight angle tells us which type of
geometric space we are in.
Page 9, # 12.
Let O denote the set of overweight people, let Q be the set of people who eat too much, let T be
the set of thrifty people, let V be the set of extravagant people, and let N be the set of Americans.
Then the four hypotheses listed in this argument are
A = People who are overweight eat too much, or O  Q
B = Overeating is an extravagance, which gives Q  V
C = Thrifty people are not extravagant, T  ~V, or taking the contrapositive, V  ~T
D = Most Americans are overweight, or N  O
If we arrange these in the order D, A, B, C, we obtain the chain N  O  Q  V  ~T, which
reduces to N  ~T, and so the valid conclusion using all these hypotheses is the statement that
Most Americans are not thrifty.
# 13. It is easiest to view this one in terms of set theory, and I will draw the picture in class.