1. Prepare a table for the finite geometry PG(2,3) using the numbers 1 to 13 for the points and showing the points on a line as one column of the table.

Solution:

1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	1
4	5	6	7	8	9	10	11	12	13		1	2
10	11	12	13	1	2	3	4	5	6	7	8	9

Exercises 3-5 refer to the above table.

3.For a set of four distinct points of the geometry, there is exactly one line of the geometry not containing any of these points.

Ans: False. Solution: Consider the four points 1,2,3 and 6. None of them appear on the lines represented by columns 4, 8 and 9. So there are three lines not containing any of these 4 points.

5. There exists a set of three distinct points in the geometry that do not have a line in common.

Ans: True. Solution:One example would be the points 1, 2 and 3. They do not appear together in any column.

7. Which of these symbols represent self-dual geometries?

• a. PG(2,4)
• b. PG(2,5)
• c. PG(2,6)
• d. PG(2,7)

9. In Fig. 1.14a, each line has how many points on it?

Ans: 4 .

11. In Fig. 1.14a, each line has how many other lines parallel to it?

Ans: None.

13. In Fig. 1.14b, each line has how many points on it?

Ans: 2.

15. In Fig. 1.14b, each line has how many other lines parallel to it?

Ans: one.

17. In Fig. 1.14c, the geometry consists of how many points and how many lines?

Ans: There are 12 points and 6 lines.

19. In Fig. 1.14c, for each two distinct points, does there exist exactly one line on both of them?

Ans: No, there are several pairs of points that have no line containing them both.