Discrete Mathematics: Elementary and Beyond

280 16. Answers to Exercises

1

1

2 3 2

4

3

4

5

5

6

6

7 7

FIGURE 16.3.

14.2 Finite Affine and Projective Planes

14.2.1. Fix any pointa. There aren+ 1 lines througha, which have no
other points in common and cover the whole plane by (a). Each of these
lines hasnpoints besidesa, so there are (n+1)npoints besidesa, and
n(n+1)+1=n^2 +n+ 1 points altogether.

14.2.2.We can assign coordinates to the vertices of the cube as if it were in
Euclidean space, but think of the coordinates as elements of the 2-element
field (Figure 16.4). Then it is straightforward (if lengthy) to check that the
planes of the Cube space are precisely the sets of points given by linear
equations. For example, the linear equationx+y+z= 1 gives the points
001 , 010 , 011 ,111 (don’t forget that we are working in the 2-element field),
which is just the plane consisting of the light points.

000

011

010

101

111

001

110

100

FIGURE 16.4.

14.2.3. A projective plane of order 10 ought to have 10^2 + 10 + 1 = 111
points, 111 lines, with 11 points on each line. The number of ways to select
a candidate line is

( 111

11

)

; the number of ways to select 111 candidate lines