 ## MT4516 Finite Mathematics

### 1.Prove that parallelism is an equivalence relation on the set of lines in an affiffiffine plane.  数学作业代写价格

2.Let i,j ∈ {1, 2, 3} be arbitrary. Prove that for each n 4 one can fifind sets P and L of points and lines so that P has n points and the axioms

(Ai) and (Aj) are satisfified.

3.Prove that in an affiffiffine plane of order n2 there are exactly n lines parallel to any given line.

### 4.A triangle in a fifinite affiffiffine plane is a set of three points not belonging to the same line. Prove that an affiffiffine plane with n2 points has exactly n 3 (n − 1) 2 (n + 1)/6 triangles.

5.A quadrangle in a fifinite affiffiffine plane is a set of four points such that no three belong to the same line. Find the number of quadrangles in  an affiffiffine plane with n 2 points.

### 6.In the affiffiffine plane AP(Z11) fifind:  数学作业代写价格

(i) an equation for the line containing the points (2, 3) and (1, 6);

(ii) an equation for the line containing the points (3, 6) and (0, 4).

(iii) the point of intersection of the lines 7x + 6y = 9 and 3x + 2y = 8.

7.In the affiffiffine plane AP(Z13) fifind:

(i) an equation for the line containing the point (3, 6) and parallel to the line 4x + 2y = 7;  数学作业代写价格

(ii) an equation for the line containing the point (1, 5) and parallel to the line 3x + 2y = 3.

8.Let F be the fifinite fifield of order pn where p is a prime difffferent from 2 and 3, and consider the affiffiffine plane AP(F). For two points (x, y),(z, t) defifine their midpoint to be the point ((x + z)/2,(y + t)/2). Let A, B, C AP(F) be non-collinear points. Prove that the three lines connecting each of these points to the midpoints of the other two all contain the same point.